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\begin{document}
\emergencystretch 3em

\begin{frontmatter}
\title{Supplementary material for: SIRUS: Making random forests interpretable}
\runtitle{Supplementary material}

\begin{aug}
\author{\fnms{Cl\'ement} \snm{B\'enard}\thanksref{m1,m2}\ead[label=e1]{clement.benard@safrangroup.com}},
\author{\fnms{G\'erard} \snm{Biau}\thanksref{m1}\ead[label=e2]{gerard.biau@upmc.fr}},
\author{\fnms{S\'ebastien} \snm{Da Veiga}\thanksref{m2}\ead[label=e3]{sebastien.da-veiga@safrangroup.com}}
\and
\author{\fnms{Erwan} \snm{Scornet}\thanksref{m3}
\ead[label=e4]{erwan.scornet@polytechnique.edu}
%\ead[label=u1,url]{http://www.foo.com}
}

%\thankstext{t1}{}
%\thankstext{t2}{}
%\thankstext{t3}{}
\runauthor{C. B\'enard, G. Biau, S. Da Veiga, E. Scornet}

\affiliation{Sorbonne Universit\'e \thanksmark{m1}, Safran Tech \thanksmark{m2} and Ecole Polytechnique \thanksmark{m3}}

\address{C. B\'enard\\
S. Da Veiga\\
Safran Tech, Modelling \& Simulation\\
Rue des Jeunes Bois \\
Châteaufort, 78114 Magny-Les-Hameaux, France\\
\printead{e1}\\
\phantom{E-mail:\ }\printead*{e3}}

\address{G. Biau\\
LPSM, Sorbonne Universit\'e\\
4 place Jussieu\\
75005 Paris, France \\
\printead{e2}}

\address{E. Scornet\\
CMAP, \'Ecole Polytechnique \\
Route de Saclay \\
91128 Palaiseau Cedex, France \\
\printead{e4}}
\end{aug}

\end{frontmatter}


\section{Proof of Theorem \ref{theorem_consistency}}
\label{sec_theorem_consistency}

We recall Assumptions (A1)-(A3) and Theorem \ref{theorem_consistency} for the sake of clarity.
\begin{enumerate}
\item[(A1)] The subsampling rate $a_n$ satisfies $\lim\limits_{n \to \infty} a_{n}=\infty$ and $\lim\limits_{n \to \infty} \frac{a_{n}}{n}=0$.
\item[(A2)] The number of trees $M_n$ satisfies $\lim\limits_{n \to \infty} M_n = \infty$.
\item[(A3)] $\bX$ has a strictly positive density $f$ with respect to the Lebesgue measure. Furthermore, for all $j\in \{1,\hdots,p\}$,
 the marginal density $f^{(j)}$ of $X^{(j)}$ is continuous, bounded, and strictly positive.
\end{enumerate}

\begin{theorem}
\label{theorem_consistency}
If Assumptions (A1)-(A3) are satisfied, then, for all $\path \in \Pi$, we have
\[
\lim\limits_{n \to \infty} \hat{p}_{M_{n},n}(\path) = p^{\star}(\path) \quad \textrm{in probability.}
\]
\end{theorem}

\subsection{Proof of Theorem \ref{theorem_consistency} for a path of one split}

First, we prove Theorem \ref{theorem_consistency} for a path of one split. The proof is extended for a path of two splits in the next subsection and follows the same steps.
Thus, we consider $\path_{1}=\{(j_{1},r_{1},s_{1})\}$ a path of one split, where $j_{1} \in\{1, \hdots, p\}$, $r_{1} \in\{1, \hdots, q - 1\}$, and $s_{1} \in\{L,R\}$.
We assume throughout that Assumptions (A1)-(A3) are satisfied. 

Before proving Theorem \ref{theorem_consistency}, we state five lemmas (Lemma \ref{convergence_quantile} to Lemma \ref{convergence_pn_theta_1}). Their proof can be found in the Subsection \ref{subsec_thm1_lemmas}.
Lemma \ref{convergence_quantile} is a preliminary technical result used to state both Lemmas \ref{criterion_residual_1} and \ref{criterion_distribution_1} - case (b).

\begin{lemme}
\label{convergence_quantile}
Let $\bX$ be a random variable distributed on $\R^p$ such that Assumptions (A1) and (A3) are satisfied. Then, for all $ j\in\{1, \hdots, p\}$ and all $r\in\{1, \hdots, q - 1\}$, we have
\[
\lim\limits_{n \to \infty} \sqrt{a_{n}}~ \mathbb{P}\big(q_{r}^{\star (j)}\leq X^{(j)}<\hat{q}_{n,r}^{(j)}\big)=0
\]
and
\[
\lim\limits_{n \to \infty} \sqrt{a_{n}}~ \mathbb{P}\big(\hat{q}_{n,r}^{(j)}\leq X^{(j)}<q_{r}^{\star (j)}\big)=0.
\]
\end{lemme}

Lemma \ref{criterion_residual_1} is used to prove both consistency (Lemma \ref{criterion_consistency_1}) and convergence rate (Lemma \ref{criterion_distribution_1})
 of the CART-splitting criterion when the root node of the tree is cut at an empirical quantile. 
Lemma \ref{convergence_pn_theta_1} is an intermediate result to prove Theorem \ref{theorem_consistency}.
\begin{lemme}
\label{criterion_residual_1}
If Assumptions (A1) and (A3) are satisfied, then for all $j\in\{1, \hdots, p\}$, all $r\in\{1, \hdots, q - 1\}$, and all $H\subseteq\mathds{R}^{p}$ such that
$\mathbb{P}(\bX \in H, X^{(j)} < q^{\star(j)}_{r})>0$ and $\mathbb{P}(\bX \in H, X^{(j)} \geq q^{\star(j)}_{r})>0$, we have
\[
\lim\limits_{n \to \infty} \sqrt{a_{n}}\big(L_{a_{n}}\big(H,\hat{q}_{n,r}^{(j)}\big)
-L_{a_{n}}\big(H,q_{r}^{\star(j)}\big)\big)=0 \quad \textrm{in probability.}
\]
\end{lemme}

\begin{lemme}
\label{criterion_consistency_1}
If Assumptions (A1) and (A3) are satisfied, then for all $j\in\{1, \hdots, p\}$, all $r\in\{1, \hdots, q - 1\}$, and all $H\subseteq\mathds{R}^{p}$ such that
$\mathbb{P}(\bX \in H, X^{(j)} < q^{\star(j)}_{r})>0$ and $\mathbb{P}(\bX \in H, X^{(j)} \geq q^{\star(j)}_{r})>0$, we have
\[
\lim\limits_{n \to \infty} L_{a_{n}}\big(H,\hat{q}_{n,r}^{(j)}\big)=L^{\star}\big(H,q_{r}^{\star(j)}\big) \quad \textrm{in probability.}
\]
\end{lemme}

When splitting a node, if the theoretical CART-splitting criterion has multiple maxima, one is randomly selected. This random selection follows a discrete probability law, which is not necessarily uniform and is based on $\P_{\bX, Y}$ as specified in Definition \ref{def_proba_equality_1}.
In order to derive the limit of the probability that a given split occurs in a $\Theta$-random tree in the empirical algorithm, one needs to assess the convergence rate of the empirical CART-splitting criterion when it has multiple maxima.

\begin{lemme} \label{criterion_distribution_1}
Consider that Assumptions (A1) and (A3) are satisfied.
Let $\bestcut_1 \subset \{1, \hdots, p\} \times \{1, \hdots, q-1\}$ be a set of splits of cardinality $c_1 \geq 2$, such that, for all $(j,r) \in \bestcut_1$,
$L^{\star}(\R^p,  q_{r}^{\star(j)}) \stackrel{\tiny{\rm def}}{=}L^{\star}_{\bestcut_1}$, i.e., the theoretical CART-splitting criterion is constant for all splits in $\bestcut_1$.
Let $(j_1, r_1) \in \bestcut_1$ and let $\textbf{L}_{n,\path_1}^{(\bestcut_1)}$ be a random vector where each component is the difference between the empirical
 CART-splitting criterion for the splits $(j, r) \in \bestcut_1 \setminus (j_1, r_1)$ and $(j_1, r_1)$, that is
\[
\textbf{L}_{n,\path_1}^{(\bestcut_1)} = \Big(
L_{a_{n}}\big(\mathbb{R}^{p},\hat{q}_{n,r}^{(j)}\big)-L_{a_{n}}\big(\mathbb{R}^{p},\hat{q}_{n,r_1}^{(j_1)}\big)\Big)_{(j, r) \in \bestcut_1 \setminus (j_1, r_1)}.
\]

\paragraph{(a)} If $L^{\star}_{\bestcut_1} > 0$, then we have
\[
\sqrt{a_{n}}~ \textbf{L}_{n,\path_1}^{(\bestcut_1)} \stackrel[n\rightarrow\infty]{\mathscr{D}}{\longrightarrow}\mathcal{N}(0, \Sigma),
\]
where, for all $(j,r), (j',r') \in \bestcut_1 \setminus (j_1, r_1)$, each element of the covariance matrix $\Sigma$ is defined by $
\Sigma_{(j,r), (j',r')} = \emph{Cov}[Z_{j,r}, Z_{j',r'}],
$ with
\begin{align*}
Z_{j,r}= \big(Y - & \E[Y | X^{(j_1)}<q_{r_1}^{\star(j_1)}]\mathds{1}_{X^{(j_1)}<q_{r_1}^{\star(j_1)}}
\\ & -\E[Y | X^{(j_1)} \geq q_{r_1}^{\star(j_1)}]\mathds{1}_{X^{(j_1)}\geq q_{r_1}^{\star(j_1)}}\big)^{2}\\
- \big(Y &- \E[Y | X^{(j)}<q_{r}^{\star(j)}]\mathds{1}_{X^{(j)}<q_{r}^{\star(j)}}
- \E[Y | X^{(j)} \geq q_{r}^{\star(j)}]\mathds{1}_{X^{(j)}\geq q_{r}^{\star(j)}}\big)^{2}.
\end{align*}
Besides, for all $(j,r) \in \bestcut_1$, $\mathds{V} [Z_{j,r}] >0$.
\paragraph{(b)} If $L^{\star}_{\bestcut_1} = 0$, then we have
\[
a_{n} \textbf{L}_{n,\path_1}^{(\bestcut_1)} \stackrel[n\rightarrow\infty]{\mathscr{D}}{\longrightarrow} h_{\path_1}(\mathbf{V}),
\]
where $\mathbf{V}$ is a Gaussian vector of covariance matrix $\emph{Cov}[\mathbf{Z}]$. If $\bestcut_1$ is explicitly written
$\bestcut_1 = \{(j_k, r_k)\}_{k=1,\hdots,c_1}$, $\mathbf{Z}$ is defined, for $k \in \{1,\hdots,c_1\}$, by
\begin{align*}
Z_{2k-1} = \frac{1}{\sqrt{p_{L,k}}} (Y - \E[Y]) \mathds{1}_{X^{(j_k)} < q_{r_k}^{\star (j_k)}} \\
Z_{2k} = \frac{1}{\sqrt{p_{R,k}}} (Y - \E[Y]) \mathds{1}_{X^{(j_k)} \geq q_{r_k}^{\star (j_k)}},
\end{align*}
where $p_{L,k} = \P (X^{(j_k)}<q_{r_k}^{\star (j_k)})$, $p_{R,k} = \P (X^{(j_k)} \geq q_{r_k}^{\star (j_k)})$, and $h_{\path_1}$ is a multivariate quadratic form defined as
\begin{align*}
h_{\path_1}: \left( \begin{array}{c} x_1 \\ \vdots \\ x_{2c_1} \end{array} \right) \rightarrow \left( \begin{array}{c}
x_{3}^2 + x_{4}^2 - x_1^2 - x_2^2\\ \vdots \\
x_{2k-1}^2 + x_{2k}^2 - x_{1}^2 - x_{2}^2 \\ \vdots \\
x_{2c_1-1}^2 + x_{2c_1}^2 - x_{1}^2 - x_{2}^2  \end{array}\right).
\end{align*}
Besides, the variance of each component of $h_{\path_1}(\mathbf{V})$ is strictly positive.
\end{lemme}

\begin{mydef}[Theoretical splitting procedure] \label{def_proba_equality_1}
Let $\theta_1^{(V)}$ be the set of eligible variables to split the root node of a theoretical random tree. 
The set of best theoretical cuts at the root node is defined as
\[
\bestcut_{1}^{\star}\big(\theta_1^{(V)}\big) = \underset{(j,r) \in \theta_{1}^{(V)} \times \{1, \hdots, q - 1\}}{\emph{argmax}} L^{\star}\big(\mathbb{R}^{p},q_{r}^{\star(j)}\big).
\]Ø
If $\bestcut_{1}^{\star}(\theta_1^{(V)})$ has multiple elements, then $(j_1, r_1)$ is randomly drawn with probability 
\begin{align} \label{def_probability_equality_1} 
\mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big) 
= \Phi_{\theta_1^{(V)}, (j_1, r_1)} (\textbf{0}),
\end{align}
where $\Phi_{\theta_1^{(V)}, (j_1, r_1)}$ is the cdf of the limit law defined in Lemma \ref{criterion_distribution_1} for 
$\bestcut_1 = \bestcut_{1}^{\star}(\theta_1^{(V)})$. 
This definition is extended for the second split in Definition \ref{def_proba_equality_2}.
\end{mydef}

Recall that the randomness in a tree can be decomposed as $\Theta = (\Theta^{(S)}, \Theta^{(V)})$, where $\Theta^{(S)}$ corresponds to the subsampling and $\Theta^{(V)}$ is related to the variable selection. $\Theta^{(V)}$ takes values in the finite set $\Omega^{(V)} = \{1,\hdots,p\}^{3 \times \texttt{mtry}}$.
\begin{lemme} \label{convergence_pn_theta_1}
If Assumptions (A1)-(A3) are satisfied, then for all $\theta^{(V)}\in\Omega^{(V)}$, we have
\[
\lim\limits_{n \to \infty} \mathbb{P}\big(\path_{1}\in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}
=\theta^{(V)}\big)=\mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big).
\]
\end{lemme}

We are now equipped to prove Theorem \ref{theorem_consistency} in the case of one single split. Recall that 
\begin{align}
    \E[\hat{p}_{M_n, n}(\path_1)] = \P (\path_1 \in T(\Theta, \mathscr{D}_{n})).
\end{align}
Since $\Theta^{(V)}$ takes values in the finite set
$\Omega^{(V)}$, according to Lemma \ref{convergence_pn_theta_1}, we have
\begin{align*}
& \lim \limits_{n \to \infty} \P (\path_1 \in T(\Theta, \mathscr{D}_{n})) \\
& \quad = 
\lim \limits_{n \to \infty} \sum_{\theta^{(V)}\in\Omega^{(V)}}\mathbb{P}\big(\path_{1}\in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}=\theta^{(V)}\big)\mathbb{P}_{\Theta^{(V)}}\big(\Theta^{(V)}=\theta^{(V)}\big)\\
& \quad = \sum_{\theta^{(V)}\in\Omega^{(V)}}\mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big)\mathbb{P}_{\Theta^{(V)}}\big(\Theta^{(V)}=\theta^{(V)}\big)\\
& \quad = \P(\path_1 \in T^{\star}(\Theta)).
\end{align*}
Therefore,
\[
\lim_{n \to \infty} \E[\hat{p}_{M_n, n}(\path_1)] = p^{\star}(\path_{1}).
\]

To finish the proof, we just have to show that 
$\lim\limits_{n \to \infty} \mathds{V}[\hat{p}_{M_{n},n}(\path_{1})] = 0$. 

The law of total variance gives
\begin{align*}
\mathbb{V} [ \hat{p}_{M_{n}, n} (\path_1)] & = \E \big[ \mathbb{V} [ \hat{p}_{M_{n}, n}(\path_1) |  \mathscr{D}_{n} ] \big] 
+  \mathbb{V} \big[ \E [ \hat{p}_{M_{n}, n} (\path_1) | \mathscr{D}_{n} ] \big] \\
& =  \E \Big[ \mathbb{V} \Big[ \frac{1}{M_{n}}\sum_{\ell=1}^{M_{n}}\mathds{1}_{\path_{1} \in T(\Theta_{\ell},\mathscr{D}_{n})}| 
 \mathscr{D}_{n} \Big] \Big]  + \mathbb{V} [ p_n (\path_1)  ] \\
\end{align*}
\begin{align}
\mathbb{V} [ \hat{p}_{M_{n}, n} (\path_1)]
 & =  \frac{1}{M_{n}} \E \big[ \mathbb{V} [ \mathds{1}_{\path_{1} \in T(\Theta_1,\mathscr{D}_{n})}| 
 \mathscr{D}_{n} ] \big] + \mathbb{V} [ p_n(\path_1)  ] \nonumber \\
 & =  \frac{1}{M_{n}} \E \big[p_n(\path_1) -p_n(\path_1)^{2} \big] + \mathbb{V} [p_n(\path_1)], \nonumber \\
& = \frac{1}{M_{n}} \E[p_n(\path_1)] ( 1 - \E[p_n(\path_1)] ) +  \Big( 1 - \frac{1}{M_{n}} \Big) \mathbb{V} [p_n(\path_1) ] . \label{proof_thm1_variance}
 \end{align}
Following the approach of \cite{mentchquantifying2016}, $p_n(\path_{1})$ 
is a complete infinite order U-statistic with the kernel $
\E [ \mathds{1}_{\path_{1} \in T ( \Theta, \mathscr{D}_{n} )} | \Theta^{( S )},  \mathscr{D}_{n} ] $. From  \cite{Hoeffding1948},
\[
\mathbb{V} [p_n(\path_1)] \leq \frac{a_{n}}{n} \xi_{a_{n}, a_{n}},
\]
where
$\xi_{a_{n}, a_{n}} = \mathbb{V} [ \E [ \mathds{1}_{\path_{1} \in T ( \Theta, \mathscr{D}_{n} )} 
| \Theta^{( S)}, \mathscr{D}_{n} ] | \Theta^{( S )} ]$. 
Since $\xi_{a_{n}, a_{n}}$ is bounded and $\lim \limits_{n \to \infty} \frac{a_{n}}{n} = 0$,
\[
\lim \limits_{n \to \infty}\mathbb{V} [p_n(\path_1)]  = 0.
\]
Using equality (\ref{proof_thm1_variance}), since $ p_n(\path_1)$ is bounded and $\lim \limits_{n \to \infty} M_{n} = \infty$,
\[
\lim \limits_{n \to \infty}\mathbb{V} [ p_{M{_n}, n}(\path_1)]  = 0.
\]
Finally, 
\begin{align*}
\lim \limits_{n \to \infty} \E\big[&(\hat{p}_{M_{n}, n} (\path_1) - p^{\star}(\path_1) )^2\big] \\ & = \lim \limits_{n \to \infty} \mathbb{V}[\hat{p}_{M_{n}, n} (\path_1)] + \big(\E[\hat{p}_{M_{n}, n} (\path_1)] - p^{\star}(\path_1)\big)^2 = 0,
\end{align*}
which concludes the proof. 


\subsection{Proof of Theorem \ref{theorem_consistency} for a path of two split}

The proof of Theorem \ref{theorem_consistency} is extended for a path of two splits. We consider $\path_{1}=\{(j_{1},r_{1},s_{1})\}$ a path of one split and
$\path_{2}=\{(j_{k},r_{k},s_{k}), k=1,2\}$ a path of
two splits, where $j_{1}, j_{2}\in\{1, ..., p\}$, $r_{1},
r_{2}\in\{1, ..., q - 1\}$ and $s_{1}, s_{2} \in\{L,R\}$.
We assume assumptions (A1)-(A3) are satisfied.

The path $\path_2 = \{ (j_1, r_1, s_1), (j_2, r_2, s_2) \}$ can occur in trees where the split at the root node is $(j_1, r_1)$
and the split of one of the child node is $(j_2, r_2)$, and in trees where the splits are made in the reversed order, $(j_2, r_2)$
at the root node and $(j_1, r_1)$ at one of the child node. Since these two events are disjoint, $\P\big(\path_{2} \in T(\Theta,\mathscr{D}_{n})\big)$
is the sum of the probability of these two events. Without loss of generality, we will consider in the entire proof that the split at the root node is
$(j_1, r_1)$. Lemmas \ref{criterion_residual_2} - \ref{convergence_pn_theta_2} below extend Lemmas \ref{criterion_residual_1} - \ref{convergence_pn_theta_1} to the case where the tree path contains two splits. 

We need to introduce additional notations, first, the theoretical hyperrectangle based on a path $\path$ by
\[
H^{\star}(\path)=\left\{ \bx \in\mathds{R}^{p}:\begin{cases}
x^{(j_{k})}<q_{r_{k}}^{\star(j_{k})} & \textrm{if }s_{k}=L\\
x^{(j_{k})}\geq q_{r_{k}}^{\star(j_{k})} & \textrm{if }s_{k}=R
\end{cases}\;,k\in 1,\hdots,d \right\},
\]
 with $d \in \{1,2\}$, the empirical counterpart of $\hat{H}_n (\path)$ defined in (2.3).
Furthermore, since from assumption (A3), $\bX$ has a strictly positive density, then for $j \in \{1, ..., p\} \setminus j_1$, and $r \in \{1, ..., q - 1\}$,
$\mathbb{P}\big(\bX \in H^{\star}(\path_{1}), X^{(j)} < q^{\star(j)}_{r}\big)>0$ and $\mathbb{P}\big(\bX \in H^{\star}(\path_{1}), X^{(j)} \geq q^{\star(j)}_{r}\big)>0$. When $j = j_1$, the second cut is performed along the same direction as the first one. 
In that case, depending on the side $s_1$ of the first cut and the cut positions $r_1$ and $r$, one of the two child node can be empty with probability one.
For example, the hyperrectangle associated to the path $\{ (1, 2, L), (1, 3, R) \}$ is empty.
In SIRUS, such splits are not considered to find the best cut for a node at the second level of the tree.
Thus we define $\bestcut_{\path_1}$ the set of possible splits for the second cut
\begin{align*}
\bestcut_{\path_1} = \{&(j,r), j \in \{1, ..., p\} \setminus j_1, r \in \{1, ..., q - 1\} \} \\
&\cup \{(j_1,r), \textrm{  s.t.  } r < r_1 \textrm{  if  } s_1 = L, \textrm{  and  } r > r_1 \textrm{  if  } s_1 = R \},
\end{align*}
and $\bestcut_{\path_1}\big(\theta_2^{(V)}\big) = \big\{(j,r) \in \bestcut_{\path_1} \textrm{  s.t.  } j \in \theta_2^{(V)}\big\}$ when the split directions are restricted to $\theta_2^{(V)} \subset \{1,...,p\}$.

\begin{lemme}
\label{criterion_residual_2}
If Assumptions (A1) and (A3) are satisfied, then for all $(j, r) \in \bestcut_{\path_1}$, we have
\begin{align*}
\lim\limits_{n \to \infty}  \sqrt{a_{n}}\big(L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)
-L_{a_{n}}\big(H^{\star}(\path_{1}),q_{r}^{\star(j)}\big)\big)=0 \quad \textrm{in probability.}
\end{align*}
\end{lemme}

\begin{lemme}
\label{criterion_consistency_2}
If Assumptions (A1) and (A3) are satisfied, then for all $(j, r) \in \bestcut_{\path_1}$, we have
\begin{align*}
\lim\limits_{n \to \infty}  L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)=L^{\star}\big(H^{\star}(\path_{1}),q_{r}^{\star(j)}\big) \quad \textrm{in probability.}
\end{align*}
\end{lemme}


\begin{lemme} \label{criterion_distribution_2}
Consider that Assumptions (A1) and (A3) are satisfied.
Let $\bestcut_1 \subset \{1, ..., p\} \times \{1, ..., q-1\}$ and $\bestcut_2 \subset \bestcut_{\path_1}$ be two sets of splits of cardinality $c_1 \geq 1$ and $c_2 \geq 2$, such that the theoretical CART-splitting criterion is constant for all splits in $\bestcut_1$ on one hand, and in $\bestcut_2$ on the other hand, i.e.,
\begin{align*}
\forall l \in \{1,2\}, \quad \forall (j,r) \in \bestcut_l, \quad L^{\star}\big(H_l,  q_{r}^{\star(j)}\big) \stackrel{\tiny{\rm def}}{=}L^{\star}_{\bestcut_l},
\end{align*}
where $H_1 = \R^p$ and $H_2 = H^{\star}(\path_1)$. Let $(j_1, r_1) \in \bestcut_1$, $(j_2, r_2) \in \bestcut_2$, and let $\textbf{L}_{n,\path_2}^{(\bestcut_1, \bestcut_2)}$ a the random vector where each component is the difference between the empirical CART-splitting criterion for the splits $(j, r) \in \bestcut_1 \setminus (j_1, r_1)$ and $(j_1, r_1)$ for the first $c_1 - 1$ components, and for the splits $(j, r) \in \bestcut_2 \setminus (j_2, r_2)$ and $(j_2, r_2)$ for the remaining $c_2 - 1$ components,  that is
\begin{align*}
\textbf{L}_{n,\path_2}^{(\bestcut_1, \bestcut_2)} = \left(\begin{array}{c}
\big[L_{a_{n}}\big(\mathbb{R}^{p},\hat{q}_{n,r}^{(j)}\big)-L_{a_{n}}\big(\mathbb{R}^{p},\hat{q}_{n,r_1}^{(j_1)}\big)\big]_{(j, r) \in \bestcut_1 \setminus (j_1, r_1)} \\
\big[L_{a_{n}}\big(\hat{H}_n(\path_{1}),\hat{q}_{n,r}^{(j)}\big)-L_{a_{n}}\big(\hat{H}_n(\path_{1}),\hat{q}_{n,r_2}^{(j_2)}\big)\big]_{(j, r) \in \bestcut_2 \setminus (j_2, r_2)} \\
\end{array}\right).
\end{align*}

\paragraph{(a)} If $L^{\star}_{\bestcut_1} > 0$ and $L^{\star}_{\bestcut_2} > 0$, then we have
\begin{align*}
\sqrt{a_{n}} \textbf{L}_{n,\path_2}^{(\bestcut_1, \bestcut_2)} \stackrel[n\rightarrow\infty]{\mathscr{D}}{\longrightarrow}\mathcal{N}(0, \Sigma)
\end{align*}
where for $l, l' \in \{1,2\}$, for all $(j,r) \in \bestcut_l \setminus (j_l,r_l)$,
$(j', r') \in \bestcut_{l'} \setminus (j_{l'}, r_{l'})$, each element of the covariance matrix $\Sigma$ is defined by
$ \Sigma_{(j,r,l), (j',r',l')} = \textrm{Cov}[Z_{j,r,l}, Z_{j',r',l'}],$ with
\begin{align*}
Z_{j,r,l}= & \frac{1}{\P(\bX \in H_l)} \big(Y - \mu_{L,r_l}^{(j_l)}\mathds{1}_{X^{(j_{l})}<q_{r_{l}}^{\star(j_{l})}}
-\mu_{R,r_l}^{(j_l)}\mathds{1}_{X^{(j_{l})}\geq q_{r_{l}}^{\star(j_{l})}}\big)^{2} \mathds{1}_{\bX \in H_l}  \\
&- \frac{1}{\P(\bX \in H_l)} \big(Y - \mu_{L,r}^{(j)}\mathds{1}_{X^{(j)}<q_{r}^{\star(j)}}
-\mu_{R,r}^{(j)}\mathds{1}_{X^{(j)}\geq q_{r}^{\star(j)}}\big)^{2} \mathds{1}_{\bX \in H_l},
\end{align*}
$\mu_{L,r}^{(j)} = \E\big[Y | X^{(j)}<q_{r}^{\star(j)}, \bX \in H_l\big]$, 
$\mu_{R,r}^{(j)} = \E\big[Y | X^{(j)} \geq q_{r}^{\star(j)}, \bX \in H_l\big]$. 
Besides, for all $l \in \{1,2\}$ and for all $(j,r) \in C_l$, $\mathds{V}[Z_{j,r,l}] > 0$.

\paragraph{(b)} If $L^{\star}_{\bestcut_1} = L^{\star}_{\bestcut_2} = 0$, then we have
\begin{align*}
a_{n} \textbf{L}_{n,\path_2}^{(\bestcut_1, \bestcut_2)} \stackrel[n\rightarrow\infty]{\mathscr{D}}{\longrightarrow} h_{\path_2}(\mathbf{V}),
\end{align*}
where $\mathbf{V}$ is a gaussian vector of covariance matrix $\textrm{Cov}[\mathbf{Z}]$. If $\bestcut_1$ and $\bestcut_2$ are explicitly written
$\bestcut_1 = \{(j_k, r_k)\}_{k \in J_1}$, and $\bestcut_2 = \{(j_k, r_k)\}_{k \in J_2}$, with $J_1 = \{1,...,c_1+1\} \setminus 2$
and $J_2 = \{2\} \cup \{c_1+2,...,c_1+c_2\}$, $\mathbf{Z}$ is defined, for $l \in \{1,2\}$ and $k \in J_l$, by
\begin{align*}
Z_{2k-1} = \frac{1}{\sqrt{p_{L,k}\P(\bX \in H_l)}} (Y - \E[Y|\bX \in H_l]) \mathds{1}_{X^{(j_k)} < q_{r_k}^{\star (j_k)}}\mathds{1}_{\bX \in H_l} \\
Z_{2k} = \frac{1}{\sqrt{p_{R,k}\P(\bX \in H_l)}} (Y - \E[Y|\bX \in H_l]) \mathds{1}_{X^{(j_k)} \geq q_{r_k}^{\star (j_k)}}\mathds{1}_{\bX \in H_l},
\end{align*}
where $p_{L,k} = \P \big(X^{(j_k)}<q_{r_k}^{\star (j_k)}, \bX \in H_l\big)$, $p_{R,k} = \P \big(X^{(j_k)} \geq q_{r_k}^{\star (j_k)}, \bX \in H_l\big)$, and $h_{\path_2}$ is a multivariate quadratic form defined as
\begin{align*}
h_{\path_2}: \left( \begin{array}{c} x_1 \\ \vdots \\ x_{2(c_1 + c_2)}  \end{array} \right) \rightarrow \left( \begin{array}{c}
x_{5}^2 + x_{6}^2 - x_1^2 - x_2^2\\ \vdots \\
x_{2c_1+1}^2 + x_{2c_1+2}^2 - x_{1}^2 - x_{2}^2\\ 
x_{2c_1+3}^2 + x_{2c_1+4}^2 - x_{3}^2 - x_{4}^2\\ \vdots \\
x_{2(c_1+c_2)-1}^2 + x_{2(c_1+c_2)}^2 - x_{3}^2 - x_{4}^2  \end{array}\right).
\end{align*}
Besides, the variance of each component of $h_{\path_2}(\mathbf{V})$ is strictly positive.

\paragraph{(c)} If $L^{\star}_{\bestcut_1} > 0$ and $L^{\star}_{\bestcut_2} = 0$, then we have
\begin{align*}
a_{n} \textbf{L}_{n,\path_2}^{(\bestcut_1, \bestcut_2)} \stackrel[n\rightarrow\infty]{\mathscr{D}}{\longrightarrow} h'_{\path_2}(\mathbf{V}),
\end{align*}
where $\mathbf{V}$ is a gaussian vector of covariance matrix $\textrm{Cov}[\mathbf{Z}]$, and $\mathbf{Z}$ is defined as,
for $k \in J_1$,
\begin{align*}
Z_{2k - 1}&=  \big( Y - \E\big[Y | X^{(j_k)}<q_{r_k}^{\star(j_k)}\big] \big)^{2} \mathds{1}_{X^{(j_{k})}<q_{r_{k}}^{\star(j_{k})}} \\
Z_{2k }&=  \big( Y - \E\big[Y | X^{(j_k)} \geq q_{r_k}^{\star(j_k)}\big] \big)^{2} \mathds{1}_{X^{(j_{k})}\geq q_{r_{k}}^{\star(j_{k})}},
\end{align*}
for $k \in J_2$,
\begin{align*}
Z_{2k - 1}&= \frac{Y - \E[Y|\bX \in H^{\star}(\path_1)]}{\sqrt{p_{L,k}\P(\bX \in H^{\star}(\path_1))}} \mathds{1}_{X^{(j_k)} < q_{r_k}^{\star (j_k)}, \bX \in H^{\star}(\path_1)} \\
Z_{2k}&= \frac{Y - \E[Y|\bX \in H^{\star}(\path_1)]}{\sqrt{p_{R,k}\P(\bX \in H^{\star}(\path_1))}} \mathds{1}_{X^{(j_k)} \geq q_{r_k}^{\star (j_k)}, \bX \in H^{\star}(\path_1)}, 
\end{align*}
and $h'_{\path_2}$ is a multivariate quadratic form defined as
\begin{align*}
h'_{\path_2}: \left( \begin{array}{c} x_1 \\ \vdots \\ x_{2(c_1 + c_2)}  \end{array} \right) \rightarrow \left( \begin{array}{c}
x_1 + x_2 - x_5 - x_6\\ \vdots \\
x_1 + x_2 - x_{2c_1+1} - x_{2c_1 + 2}\\ 
x_{2c_1+3}^2 + x_{2c_1+4}^2 - x_{3}^2 - x_{4}^2\\ \vdots \\
x_{2(c_1+c_2)-1}^2 + x_{2(c_1+c_2)}^2 - x_{3}^2 - x_{4}^2  \end{array}\right).
\end{align*}
Besides, the variance of each component of $h'_{\path_2}(\mathbf{V})$ is strictly positive.

\paragraph{(d)} $L^{\star}_{\bestcut_1} = 0$ and $L^{\star}_{\bestcut_2} > 0$. Symmetric to case (c).
\end{lemme}


\begin{mydef}[Theoretical splitting procedure at children nodes] \label{def_proba_equality_2}
Let $\theta^{(V)} = (\theta_1^{(V)}, \theta_2^{(V)}, \cdot) \in \Omega^{(V)}$ be the sets of eligible variables to split the nodes of a theoretical random tree. The set of best theoretical cuts at the left children node along the variables in $\theta^{(V)}_2$ is defined as  
\begin{align*}
\bestcut_{2}^{\star}\big(\theta_{2}^{(V)}\big) = \underset{ (j,r) \in \bestcut_{\path_1}\big(\theta_2^{(V)}\big)}{\textrm{argmax}}
 L^{\star}\big(H^{\star}(\path_{1}),q_{r}^{\star(j)}\big).    
\end{align*}
If $\bestcut_{2}^{\star}\big(\theta_2^{(V)}\big)$ has multiple elements, then $(j_2, r_2)$ is randomly drawn with probability 
\begin{align} \label{def_probability_equality_2} 
\mathbb{P}\big(\path_{2} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big) 
= \frac{\Phi_{\path_1, \theta^{(V)}, (j_2, r_2)} (\textbf{0})}
{\mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big)},
\end{align}
where $\mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big)$ is defined
from Definition \ref{def_proba_equality_1}, and $\Phi_{\path_1, \theta^{(V)}, (j_2, r_2)}$ is the cdf of the limit law defined in Lemma \ref{criterion_distribution_2} for $\bestcut_1 = \bestcut_{1}^{\star}\big(\theta_1^{(V)}\big)$ and
$\bestcut_2 = \bestcut_{2}^{\star}\big(\theta_2^{(V)}\big)$.
\end{mydef}

\begin{lemme}
\label{convergence_pn_theta_2}
If Assumptions (A1)-(A3) are satisfied, then for all $\theta^{(V)}\in\Omega^{(V)}$, we have
\begin{align*}
\lim\limits_{n \to \infty} \mathbb{P}\big(\path_{2} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}
=\theta^{(V)}\big)=\mathbb{P}\big(\path_{2} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big)
\end{align*}
\end{lemme}
Finally, the proof of Theorem \ref{theorem_consistency} in the two-splits scenario is the same as in the single-split scenario. 




\subsection{Proofs of intermediate lemmas} \label{subsec_thm1_lemmas}

\begin{proof}[Proof of Lemma \ref{convergence_quantile}]
\label{proof_lemma_convergence_quantile}

Set $j\in\{1, ..., p\} $, and $r\in\{1, ..., q - 1\}$. We define  the marginal cumulative distribution function $F^{(j)}$ of $X^{(j)}$,
 $F^{(j)}(x)=\mathbb{P}\big(X^{(j)}<x\big)$, and $F_{n}^{(j)}$ the empirical c.d.f.
\begin{align*}
F_{n}^{(j)}(x)=\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{X^{(j)}_{i}\leq x}.
\end{align*}
We adapt an inequality from \cite{serfling2009approximation} (section 2.3.2 page 75)
to bound the following conditional probability for all $\varepsilon>0$
\begin{align}
\P\big(q_{r}^{\star(j)}&\leq X_{1}^{(j)} < \hat{q}_{n,r}^{(j)} | X_1^{(j)} = q_{r}^{\star(j)} + \varepsilon \big) \nonumber \\
=\P\big(&q_{r}^{\star(j)} + \varepsilon < \hat{q}_{n,r}^{(j)} | X_1^{(j)} = q_{r}^{\star(j)} + \varepsilon \big) \nonumber \\
\leq \P\big(& F_{n}^{(j)}\big(q_{r}^{\star(j)} + \varepsilon \big)
\leq F_{n}^{(j)}\big(\hat{q}_{n,r}^{(j)}\big) | X_1^{(j)} = q_{r}^{\star(j)} + \varepsilon \big) \nonumber \\
\leq \mathbb{P}\Big(&1 + \sum_{i=2}^{n}\mathds{1}_{X_{i}^{(j)}\leq q_{r}^{\star(j)} + \varepsilon}
\leq\left\lceil \frac{n.r}{q}\right\rceil \Big) \nonumber \\
\leq \mathbb{P}\Big(&\sum_{i=2}^{n}\mathds{1}_{X_{i}^{(j)}\leq q_{r}^{\star(j)} + \varepsilon}
- (n - 1) F^{(j)}\big(q_{r}^{\star(j)} + \varepsilon \big) \\
&\leq\left\lceil \frac{n.r}{q}\right\rceil - 1 - (n - 1) F^{(j)}\big(q_{r}^{\star(j)} + \varepsilon\big)\Big) \label{eq_hoeff}
\end{align}

Since $f$ is continuous and strictly positive, there exists three constants $c_1, c_2, \eta >0$ such that for all $x \in [q_{r}^{\star(j)}, q_{r}^{\star(j)} + \eta]$, 
$c_1 \leq f^{(j)}(x) \leq c_2$. Thus, for all $\varepsilon < \eta$, we have
\begin{align*}
 F^{(j)}\big(q_{r}^{\star(j)} + \varepsilon \big) - F^{(j)}\big(q_{r}^{\star(j)} \big)  & =  \int_{q_{r}^{\star(j)} }^{q_{r}^{\star(j)} + \varepsilon} f^{(j)}(x) \textrm{d} x,
\end{align*}
which leads to
\begin{align*}
c_1 \varepsilon \leq F^{(j)}\big(q_{r}^{\star(j)} + \varepsilon \big) - F^{(j)}\big(q_{r}^{\star(j)} \big) \leq c_2 \varepsilon.
\end{align*}
Consequently,
\begin{align*}
\left\lceil \frac{n.r}{q}\right\rceil - 1 - &(n - 1) F^{(j)}\big(q_{r}^{\star(j)} + \varepsilon\big) \\
& \leq \left\lceil \frac{n.r}{q}\right\rceil - 1 - (n - 1) \big( c_1 \varepsilon + F^{(j)}\big(q_{r}^{\star(j)}\big) \big) \\
& \leq \left\lceil \frac{n.r}{q}\right\rceil - 1 - (n - 1) c_1 \varepsilon -  \frac{(n-1).r}{q} \\
& \leq 1 - (n - 1) c_1 \varepsilon.
\end{align*}
For $n > 1 + \frac{1}{c_1 \varepsilon}$, we can apply Hoeffding inequality to \ref{eq_hoeff}, 
\begin{align}
\P\big(&q_{r}^{\star(j)}\leq X_{1}^{(j)} < \hat{q}_{n,r}^{(j)} | X_1^{(j)} = q_{r}^{\star(j)} + \varepsilon \big) \nonumber \\
& \leq \mathbb{P}\big(\sum_{i=2}^{n}\mathds{1}_{X_{i}^{(j)}\leq q_{r}^{\star(j)} + \varepsilon}
- (n - 1) F^{(j)}\big(q_{r}^{\star(j)} + \varepsilon \big)
\leq 1 - (n - 1) c_1 \varepsilon \big) \nonumber \\
& \leq e^{-\frac{2}{n}\big(1 - (n - 1) c_1  \varepsilon \big)^{2}} \nonumber \\
& \leq C e^{- 2 n c_1^{2}  \varepsilon^{2}}, \label{bound_qtl}
\end{align}
where $C = e^{2 c_1 \eta (1 + 2 c_1 \eta)}$. By definition, we have
\begin{align*}
\P\big(q_{r}^{\star(j)}\leq X_{1}^{(j)} < \hat{q}_{n,r}^{(j)})
= \int_{]0,\infty[}\P\big(&q_{r}^{\star(j)}\leq X_{1}^{(j)} < \hat{q}_{n,r}^{(j)} | X_1^{(j)} = q_{r}^{\star(j)} + \varepsilon \big) \\
& \times f^{(j)}\big(q_{r}^{\star(j)} + \varepsilon\big) d\varepsilon.
\end{align*}
To bound the previous integral, we break it down in three parts. Since $f^{(j)}$ is bounded by $c_2$ on $[q_{r}^{\star(j)}, q_{r}^{\star(j)} + \eta]$,
 for $n > 1 + \frac{1}{c_1 \eta}$ we use inequality \ref{bound_qtl} to get
\begin{align*}
\P\big(q_{r}^{\star(j)}\leq X_{1}^{(j)} < \hat{q}_{n,r}^{(j)}\big)
\leq &\int_{]0,\frac{1}{(n - 1) c_1}]} c_2 d\varepsilon \\
&+ \int_{]\frac{1}{(n - 1) c_1}, \eta[} c_2 C e^{- 2 n c_1^{2}  \varepsilon^{2}}  d\varepsilon \\
&+ \int_{[\eta, \infty[}C e^{- 2 n c_1^{2}  \eta^{2}} f^{(j)}\big(q_{r}^{\star(j)} + \varepsilon\big)  d\varepsilon. \\
\end{align*}
In the second integral, we introduce the following change of variable $u =\sqrt{2n}c_1\varepsilon$
\begin{align*}
\int_{]\frac{1}{(n - 1) c_1}, \eta[} c_2 C e^{- 2 n c_1^{2}  \varepsilon^{2}}  d\varepsilon 
&= \frac{c_2 C}{c_1 \sqrt{2n}} \int_{]\frac{\sqrt{2n}}{(n - 1)}, \sqrt{2n} c_ 1 \eta[} e^{- u^2}  du \\
&\leq \frac{c_2 C}{c_1 \sqrt{2n}} \int_{]0,\infty[} e^{- u^2}  du 
\leq \frac{\sqrt{\pi} c_2 C}{2 c_1 \sqrt{2n}},
 \end{align*}
and therefore we can write
\begin{align*}
\sqrt{a_n} \P\big(q_{r}^{\star(j)}\leq X_{1}^{(j)} < \hat{q}_{n,r}^{(j)}\big)
\leq \frac{c_2 \sqrt{a_n} }{(n - 1) c_1} + \frac{\sqrt{\pi a_n } c_2 C}{2 c_1 \sqrt{2n}}
+ C \sqrt{a_n}  e^{- 2 n c_1^{2}  \eta^{2}}
\end{align*}

From Assumption (A1), $\lim \limits_{n \to \infty} \frac{a_n}{n} = 0$, and then
$$
\lim_{n \to \infty} \sqrt{a_n}~ \mathbb{P}\big(q_{r}^{\star(j)}\leq X_{1}^{(j)}<\hat{q}_{n,r}^{(j)}\big)  = 0.
$$
The case $\lim \limits_{n \to \infty} \sqrt{a_{n}} \mathbb{P}\big(\hat{q}_{n,r}^{(j)}\leq X_{1}^{(j)}
<q_{r}^{\star(j)}\big) = 0$ is similar.
\end{proof}

\newpage
\subsubsection{Case 1: $\path_{1}$}

\begin{proof}[Proof of Lemma \ref{criterion_residual_1}]
Let $j\in\{1, ..., p\} $, $r\in\{1, ..., q - 1\} $,
and $H\subseteq\mathds{R}^{p}$ such that $\mathbb{P}\big(\bX \in H, X^{(j)} < q^{\star(j)}_{r}\big)>0$
 and $\mathbb{P}\big(\bX \in H, X^{(j)} \geq q^{\star(j)}_{r}\big)>0$. Let
 
\begin{align*}
 \Delta_{n,r}^{(j)} & =\sqrt{a_{n}}\big(L_{a_{n}}\big(H,\hat{q}_{n,r}^{(j)}\big)-L_{a_{n}}\big(H,q_{r}^{\star(j)}\big)\big) 
\end{align*}
that is
\begin{align*}
 \Delta_{n,r}^{(j)}  & =-\frac{\sqrt{a_{n}}}{N_{n}(H)}\big[\sum_{i=1}^{a_{n}}\big(Y_{i}-\overline{Y}_{H_{L}}\mathds{1}_{X_{i}^{(j)}<\hat{q}_{n,r}^{(j)}}-\overline{Y}_{H_{R}}\mathds{1}_{X_{i}^{(j)}\geq\hat{q}_{n,r}^{(j)}}\big)^{2}\mathds{1}_{\bX_{i}\in H}\\
& \quad -\sum_{i=1}^{a_{n}}\big(Y_{i}-\overline{Y}_{H_{L}^{\star}}\mathds{1}_{X_{i}^{(j)}<q_{r}^{\star(j)}}-\overline{Y}_{H_{R}^{\star}}\mathds{1}_{X_{i}^{(j)}\geq q_{r}^{\star(j)}}\big)^{2}\mathds{1}_{\bX_{i}\in H}\big]
\end{align*}
where, for a generic hyperrectangle $H$, we define $N_n(H) = \sum_{i=1}^{a_n} \mathds{1}_{\bX_i \in H}$,  and
\begin{align*}
H_{L}=\big\{ \bx \in H:x^{(j)}<\hat{q}_{n,r}^{(j)}\big\} \quad \textrm{and} \quad  \overline{Y}_{H_{L}}=\frac{1}{N_{n}(H_{L})}\sum_{i=1}^{a_{n}}Y_{i}\mathds{1}_{X_{i}^{(j)}<\hat{q}_{n,r}^{(j)}}\mathds{1}_{\bX_{i}\in H},
\end{align*}
with the convention $\overline{Y}_{H_L} = 0$ if $H_L$ is empty.
The theoretical quantities $H_L^{\star}$ and  $\overline{Y}_{H_{L}^{\star}}$ are
defined similarly by replacing the empirical quantile by its population version. We define symmetrically $H_R$, $H_R^{\star}$, $\overline{Y}_{H_{R}}$,
$\overline{Y}_{H_{R}^{\star}}$. 

Simple calculations show that \begin{align}
\Delta_{n,r}^{(j)}& =\frac{\sqrt{a_{n}}}{N_{n}(H)}\big(\overline{Y}_{H_{L}}^{2}N_{n}(H_{L})-\overline{Y}_{H_{L}^{\star}}^{2}N_{n}(H_{L}^{\star})\big) \nonumber \\
& \quad +\frac{\sqrt{a_{n}}}{N_{n}(H)}\big(\overline{Y}_{H_{R}}^{2}N_{n}(H_{R})-\overline{Y}_{H_{R}^{\star}}^{2}N_{n}(H_{R}^{\star})\big)
\label{proof_lemme2_eq1}
\end{align}

The first term in equation (\ref{proof_lemme2_eq1}) can be rewritten as
\begin{align*}
&\frac{\sqrt{a_{n}}}{N_{n}(H\big)}(\overline{Y}_{H_{L}}^{2}N_{n}(H_{L})-\overline{Y}_{H_{L}^{\star}}^{2}N_{n}(H_{L}^{\star})\big)\\
& \quad =\frac{\sqrt{a_{n}}}{N_{n}(H)N_{n}(H_{L})N_{n}(H_{L}^{\star})}\sum_{i,k,l=1}^{a_{n}}Y_{i}Y_{k} 
\mathds{1}_{\bX_{i}\in H,\bX_{k}\in H} \\ & \qquad \times \big(\mathds{1}_{X_{l}^{(j)}<q_{r}^{\star(j)}}\mathds{1}_{X_{i}^{(j)}<\hat{q}_{n,r}^{(j)}}\mathds{1}_{X_{k}^{(j)}<\hat{q}_{n,r}^{(j)}} - \mathds{1}_{X_{l}^{(j)}<\hat{q}_{n,r}^{(j)}}\mathds{1}_{X_{i}^{(j)}<q_{r}^{\star(j)}}\mathds{1}_{X_{k}^{(j)}<q_{r}^{\star(j)}}\big).
\end{align*}
Since $Y_{i}\in \{ 0,1\}$, we have the following bound

\begin{align*}
& \frac{\sqrt{a_{n}}}{N_{n}(H)}\big|\overline{Y}_{H_{L}}^{2}N_{n}(H_{L})-\overline{Y}_{H_{L}^{\star}}^{2}N_{n}(H_{L}^{\star})\big| \nonumber \\
&  \leq\frac{\sqrt{a_{n}}}{N_{n}(H)N_{n}(H_{L})N_{n}(H_{L}^{\star})}\sum_{i,k,l=1}^{a_{n}}\big|\mathds{1}_{X_{l}^{(j)}<q_{r}^{\star(j)}}\mathds{1}_{X_{i}^{(j)}<\hat{q}_{n,r}^{(j)}}\mathds{1}_{X_{k}^{(j)}<\hat{q}_{n,r}^{(j)}} \\
& \quad -\mathds{1}_{X_{l}^{(j)}<\hat{q}_{n,r}^{(j)}}\mathds{1}_{X_{i}^{(j)}<q_{r}^{\star(j)}}\mathds{1}_{X_{k}^{(j)}<q_{r}^{\star(j)}}\big|,
\end{align*}
and finally
\begin{align}
\label{delta_bound_1}
\frac{\sqrt{a_{n}}}{N_{n}(H)}\big|\overline{Y}_{H_{L}}^{2}N_{n}(H_{L})-\overline{Y}_{H_{L}^{\star}}^{2}N_{n}(H_{L}^{\star})\big|
& \leq\frac{a_{n}^{3}}{N_{n}(H)N_{n}(H_{L})N_{n}(H_{L}^{\star})}W_{n,r}^{(j)},
\end{align}
where
\begin{align} \label{proof_lemma2_eq2_W}
W_{n,r}^{(j)}=\frac{\sqrt{a_{n}}}{a_{n}^{3}}\sum_{i,k,l=1}^{a_{n}}\big|&\mathds{1}_{X_{l}^{(j)}<q_{r}^{\star(j)}}\mathds{1}_{X_{i}^{(j)}<\hat{q}_{n,r}^{(j)}}\mathds{1}_{X_{k}^{(j)}<\hat{q}_{n,r}^{(j)}}\\&-\mathds{1}_{X_{l}^{(j)}<\hat{q}_{n,r}^{(j)}}\mathds{1}_{X_{i}^{(j)}<q_{r}^{\star(j)}}\mathds{1}_{X_{k}^{(j)}<q_{r}^{\star(j)}}\big|. \nonumber
\end{align}
A close inspection of the terms inside the sum of (\ref{proof_lemma2_eq2_W}) reveals that 
\begin{align*}
\mathbb{E}\big[W_{n,r}^{(j)}\big] & \leq\frac{\sqrt{a_{n}}}{a_{n}^{3}}\sum_{i,k,l=1}^{a_{n}}\mathbb{P}\big(\hat{q}_{n,r}^{(j)}\leq X_{i}^{(j)}<q_{r}^{\star(j)}\big)+\mathbb{P}\big(\hat{q}_{n,r}^{(j)}\leq X_{k}^{(j)}<q_{r}^{\star(j)}\big)\\
& \quad +\mathbb{P}\big(q_{r}^{\star(j)}\leq X_{l}^{(j)}<\hat{q}_{n,r}^{(j)}\big)+\mathbb{P}\big(q_{r}^{\star(j)}\leq X_{i}^{(j)}<\hat{q}_{n,r}^{(j)}\big)\\
& \quad +\mathbb{P}\big(q_{r}^{\star(j)}\leq X_{k}^{(j)}<\hat{q}_{n,r}^{(j)}\big)+\mathbb{P}\big(\hat{q}_{n,r}^{(j)}\leq X_{l}^{(j)}<q_{r}^{\star(j)}\big)\\
& \leq 3\sqrt{a_{n}}~\mathbb{P}\big(\hat{q}_{n,r}^{(j)}\leq X_{1}^{(j)}<q_{r}^{\star(j)}\big)+3\sqrt{a_{n}}~\mathbb{P}\big(q_{r}^{\star(j)}\leq X_{1}^{(j)}<\hat{q}_{n,r}^{(j)}\big),
\end{align*}
which tends to zero, according to Lemma \ref{convergence_quantile}. Thus, in probability, 
\begin{align}
\lim\limits_{n \to \infty} W_{n,r}^{(j)} = 0. \label{proof_lemma2_eq:1}
\end{align}
Regarding the remaining terms in inequality (\ref{delta_bound_1}), by the law of large numbers, in probability,   
\begin{align}
\lim\limits_{n \to \infty} \frac{N_{n}(H)}{a_{n}} = \mathbb{P}\big(\bX\in H\big), \quad
\lim\limits_{n \to \infty} \frac{N_{n}(H_{L}^{\star})}{a_{n}} = \mathbb{P}\big(\bX\in H_{L}^{\star}\big).
\label{proof_lemma2_eq:2}
\end{align}
Additionally, 
\begin{align*}
\mathbb{E}\big[\big|\frac{N_{n}(H_{L})}{a_{n}}-\frac{N_{n}(H_{L}^{\star})}{a_{n}}\big|\big] & \leq \mathbb{E}\big[ \frac{1}{a_{n}}\sum_{i=1}^{a_{n}}\mathds{1}_{X^{(j)}_{i}\in H}\big| \mathds{1}_{X^{(j)}_{i}\leq\hat{q}_{n,r}^{(j)}}-\mathds{1}_{X^{(j)}_{i}\leq q_{r}^{\star(j)}}\big|\big]\\
& \leq \mathbb{P}\big(\hat{q}_{n,r}^{(j)}\leq X_{1}^{(j)}<q_{r}^{\star(j)}\big)+\mathbb{P}\big(q_{r}^{\star(j)}\leq X_{1}^{(j)}<\hat{q}_{n,r}^{(j)}\big),
\end{align*}
which tends to zero, according to Lemma \ref{convergence_quantile}. Therefore, in probability, 
\begin{align}
\lim\limits_{n \to \infty} \frac{N_{n}(H_{L})}{a_{n}}-\frac{N_{n}(H_{L}^{\star})}{a_{n}} = 0.
\label{proof_lemma2_eq:3}    
\end{align}
Since $\mathbb{P}(\bX\in H) > 0$ and $\mathbb{P}(\bX\in H_{L}^{\star}) > 0$ by assumption, 
we can combine (\ref{proof_lemma2_eq:1})-(\ref{proof_lemma2_eq:3}) to obtain, in probability, 
\begin{align}
\lim\limits_{n \to \infty} \frac{a_{n}^{3}}{N_{n}(H)N_{n}(H_{L})N_{n}(H_{L}^{\star})}=\frac{1}{\mathbb{P}
(\bX\in H)\mathbb{P}(\bX\in H_{L}^{\star})^{2}}. \label{eq:2}
\end{align}
Using (\ref{proof_lemma2_eq:1}) and (\ref{eq:2}) and inequality (\ref{delta_bound_1}), we obtain, in probability, 
\begin{align*}
 \lim\limits_{n \to \infty} \frac{\sqrt{a_{n}}}{N_{n}(H)}\big|\overline{Y}_{H_{L}}^{2}N_{n}(H_{L})-\overline{Y}_{H_{L}^{\star}}^{2}N_{n}(H_{L}^{\star})\big| = 0.
 \end{align*}
Similar results can be derived for the other term in equation (\ref{proof_lemme2_eq1}), which allows us to conclude that, in probability, 
\begin{align*}
 \lim\limits_{n \to \infty} \sqrt{a_{n}} \big(L_{a_{n}}\big(H,\hat{q}_{n,r}^{(j)}\big)-L_{a_{n}}\big(H,q_{r}^{\star(j)}\big)\big) = 0.
\end{align*}
\end{proof}



\begin{proof}[Proof of Lemma \ref{criterion_consistency_1}]

Let $j\in\{1, ..., p\}$, $r\in\{1, ..., q - 1\}$ and $H\subseteq\mathds{R}^{p}$ such that 
$\mathbb{P}\big(\bX \in H, X^{(j)} < q^{\star(j)}_{r}\big)>0$ and $\mathbb{P}\big(\bX \in H, X^{(j)} \geq q^{\star(j)}_{r}\big)>0$.
\begin{align*}
L_{a_{n}}\big(H,\hat{q}_{n,r}^{(j)}\big) = L_{a_{n}}\big(H,q_{r}^{\star(j)}\big) +
\big(L_{a_{n}}\big(H,\hat{q}_{n,r}^{(j)}\big) - L_{a_{n}}\big(H,q_{r}^{\star(j)}\big)\big)
\end{align*}
From the law of large number, in probability, 
\begin{align*}
\lim \limits_{n \to \infty} L_{a_{n}}\big(H,q_{r}^{\star(j)}\big)
= L^{\star}\big(H,q_{r}^{\star(j)}\big).
\end{align*}
Thus, according to Lemma \ref{criterion_residual_1}, in probability, 
\begin{align*}
\lim \limits_{n \to \infty} L_{a_{n}}\big(H,\hat{q}_{n,r}^{(j)}\big)
= L^{\star}\big(H,q_{r}^{\star(j)}\big).
\end{align*}
\end{proof}



\begin{proof}[Proof of Lemma \ref{criterion_distribution_1}]
We consider $\bestcut_1$, a set of splits of cardinality $c_1 \geq 2$ satisfying, for all $(j,r) \in \bestcut_1$,
$L^{\star}\big(\R^p,  q_{r}^{\star(j)}\big) \stackrel{\tiny{\rm def}}{=}L^{\star}_{\bestcut_1}$.
Fix $(j_1, r_1) \in \bestcut_1$, we recall that
\begin{align*}
\textbf{L}_{n,\path_1}^{(\bestcut_1)} = \left(\begin{array}{c}
L_{a_{n}}\big(\mathbb{R}^{p},\hat{q}_{n,r}^{(j)}\big)-L_{a_{n}}\big(\mathbb{R}^{p},\hat{q}_{n,r_1}^{(j_1)}\big)
\end{array}\right)_{(j, r) \in \bestcut_1 \setminus (j_1, r_1)}.
\end{align*}

\paragraph{Case (a): $L^{\star}_{\bestcut_1} > 0$} 
We first consider the following decomposition for $(j,r) \in \bestcut_1$,
\begin{align*}
L_{a_{n}}\big(&\R^p,\hat{q}_{n,r}^{(j)}\big) = L_{a_{n}}\big(\R^p,q_{r}^{\star(j)}\big) +
\big(L_{a_{n}}\big(\R^p,\hat{q}_{n,r}^{(j)}\big) - L_{a_{n}}\big(\R^p,q_{r}^{\star(j)}\big)\big) \\
=& \frac{1}{a_{n}}\sum_{i=1}^{n}(Y_{i}-\overline{Y})^{2}
-\frac{1}{a_{n}}\sum_{i=1}^{n}\big(Y_{i}-\overline{Y}^{\star}_{L}\mathds{1}_{X_{i}^{(j)}<q_{r}^{\star(j)}}
-\overline{Y}^{\star}_{R}\mathds{1}_{X_{i}^{(j)}\geq q_{r}^{\star(j)}}\big)^{2} \\
&+ L_{a_{n}}\big(\R^p,\hat{q}_{n,r}^{(j)}\big) - L_{a_{n}}\big(\R^p,q_{r}^{\star(j)}\big), 
\end{align*}
where 
\begin{align*}
N_{n,L}^{\star} = \sum_{i=1}^{a_{n}} \mathds{1}_{X_{i}^{(j)}<q_{r}^{\star(j)}}    \quad \textrm{and} \quad \overline{Y}_{L}^{\star}=\frac{1}{N_{n,L}^{\star}} \sum_{i=1}^{a_{n}}Y_{i}\mathds{1}_{X_{i}^{(j)}<q_{r}^{\star(j)}}
\end{align*}
($\overline{Y}_{R}^{\star}$, $N_{n,R}^{\star}$ are defined symmetrically).
Letting $\mu_{L,r}^{(j)} = \E\big[Y | X^{(j)}<q_{r}^{\star(j)}\big]$ (and $\mu_{R,r}^{(j)}$ symmetrically), the first two terms of the last decomposition are standard variance estimates and we can write
\begin{align}
L_{a_{n}}\big(\R^p,\hat{q}_{n,r}^{(j)}\big) 
= &\frac{1}{a_{n}}\sum_{i=1}^{n}(Y_{i}-\overline{Y})^{2} \\
&-\frac{1}{a_{n}}\sum_{i=1}^{n}\big(Y_{i}- \mu_{L,r}^{(j)}\mathds{1}_{X_{i}^{(j)}<q_{r}^{\star(j)}}
-\mu_{R,r}^{(j)}\mathds{1}_{X_{i}^{(j)}\geq q_{r}^{\star(j)}}\big)^{2} + R_{n,r}^{(j)}, \label{eq_lemma4_1}
\end{align}
where
\begin{align}  \label{eq_lemma4}
R_{n,L}^{(j)} =& \frac{N_{n,L}^{\star}}{a_{n}} \big(\overline{Y}^{\star}_{L} - \mu_{L,r}^{(j)}\big)^2 
+ \frac{N_{n,R}^{\star}}{a_{n}}\big(\overline{Y}^{\star}_{R} - \mu_{L,r}^{(j)}\big)^{2} \\ &+ L_{a_{n}}\big(\R^p,\hat{q}_{n,r}^{(j)}\big)
 - L_{a_{n}}\big(\R^p,q_{r}^{\star(j)}\big). \nonumber
\end{align}
Using the Central limit theorem, in probability,
\begin{align} \label{eq_proof_normal_1}
\lim \limits_{n \to \infty} \sqrt{a_n}~ \frac{N_{L,r}^{\star}}{a_{n}} \big(\overline{Y}^{\star}_{L,r} - \mu_{L,r}^{(j)}\big)^2 = 0.
\end{align}
The same result holds for the second term of (\ref{eq_lemma4}), and using Lemma \ref{criterion_residual_1} for the third term of (\ref{eq_lemma4}),
we get that, in probability,
\begin{align*}
\lim \limits_{n \to \infty} \sqrt{a_n} \big(L_{a_{n}}\big(\R^p,\hat{q}_{n,r}^{(j)}\big) - L_{a_{n}}\big(\R^p,q_{r}^{\star(j)}\big) \big) = 0.
\end{align*}
Finally,
\begin{align*}
\lim \limits_{n \to \infty} \sqrt{a_n} R_{n,r}^{(j)} = 0, \quad \textrm{in probability}.
\end{align*}
%
Using Equation (\ref{eq_lemma4_1}), each component of $\textbf{L}_{n,\path_1}^{(\bestcut_1)}$ writes, with $(j,r) \in \bestcut_1 \setminus (j_1,r_1)$,
\begin{align*}
L_{a_{n}}\big(\R^p,\hat{q}_{n,r}^{(j)}\big)  &- L_{a_{n}}\big(\R^p,\hat{q}_{n,r_1}^{(j_1)}\big) \\
=& \frac{1}{a_{n}}\sum_{i=1}^{n}\big(Y_{i} - \mu_{L,r_1}^{(j_1)}\mathds{1}_{X_{i}^{(j_1)}<q_{r_1}^{\star(j_1)}}
-\mu_{R,r_1}^{(j_1)}\mathds{1}_{X_{i}^{(j_1)}\geq q_{r_1}^{\star(j_1)}}\big)^{2} \\
&- \big(Y_{i} - \mu_{L,r}^{(j)}\mathds{1}_{X_{i}^{(j)}<q_{r}^{\star(j)}}
-\mu_{R,r}^{(j)}\mathds{1}_{X_{i}^{(j)}\geq q_{r}^{\star(j)}}\big)^{2} \\ &+ R_{n,r}^{(j)} - R_{n,r_1}^{(j_1)}
 \end{align*}
%
We can apply the multivariate Central limit theorem and Slutsky's theorem to obtain, 
\begin{align*}
\sqrt{a_{n}}~ \textbf{L}_{n,\path_1}^{(\bestcut_1)} \stackrel[n\rightarrow\infty]{\mathscr{D}}{\longrightarrow}\mathcal{N}\big(0, \Sigma\big)
\end{align*}
where for all $(j,r), (j',r') \in \bestcut_1 \setminus (j_1, r_1)$, each element of the covariance matrix $\Sigma$ is defined by $
\Sigma_{(j,r), (j',r')} = \textrm{Cov}[Z_{j,r}, Z_{j',r'}],
$ with
\begin{align*}
Z_{j,r}= & \big(Y - \mu_{L,r_1}^{(j_1)}\mathds{1}_{X^{(j_1)}<q_{r_1}^{\star(j_1)}}
-\mu_{R,r_1}^{(j_1)}\mathds{1}_{X^{(j_1)}\geq q_{r_1}^{\star(j_1)}}\big)^{2}\\
&- \big(Y - \mu_{L,r}^{(j)}\mathds{1}_{X^{(j)}<q_{r}^{\star(j)}}
-\mu_{R,r}^{(j)}\mathds{1}_{X^{(j)}\geq q_{r}^{\star(j)}}\big)^{2}.
\end{align*}
Since $L^{\star}_{\bestcut_1} > 0$, we have for all $(j,r) \in \bestcut_1$, $\mu_{L,r}^{(j)} \neq \mu_{R,r}^{(j)}$. Besides, according to assumption (A3), $\bX$ has a strictly positive density. Consequently, the variance of $Z_{j,r}$ is strictly positive. This concludes the first case. 


\paragraph{Case (b): $L^{\star}_{\bestcut_1} = 0$}
Fix $(j,r) \in \bestcut_1$. Since $L^{\star}\big(\R^p, q_{r}^{\star(j)}\big) = 0$, we have 
\begin{align*}
\E[Y] = \E\big[Y|X^{(j)} < q^{\star (j)}_{r}\big] = \E\big[Y|X^{(j)} \geq q^{\star (j)}_{r}\big] \stackrel{\tiny{\rm def}}{=} \mu. 
\end{align*}
Then, simple calculations show that $L_{a_{n}}\big(\mathbb{R}^{p},\hat{q}_{n,r}^{(j)}\big)$ writes
\begin{align*}
L_{a_{n}}\big(\mathbb{R}^{p},\hat{q}_{n,r}^{(j)}\big) = - (\overline{Y} - \mu)^2 + \underbrace{\frac{N_{n,L}}{a_n}(\overline{Y}_{L} - \mu)^2}_{\delta_{L}} 
 + \underbrace{\frac{N_{n,R}}{a_n}(\overline{Y}_{R} - \mu)^2}_{\delta_{R}},
\end{align*}
where 
\begin{align*}
N_{n,L} = \sum_{i=1}^{a_n} \mathds{1}_{X_i^{(j)} < \hat{q}_{n,r}^{(j)}}  
\quad \textrm{and}
\quad 
\overline{Y}_{L}=\frac{1}{N_{n,L}}\sum_{i=1}^{a_{n}}Y_{i}\mathds{1}_{X_{i}^{(j)}<\hat{q}_{n,r}^{(j)}}
\end{align*}
($N_{n,R}$, $\overline{Y}_{R}$ are defined similarly for the other cell).
Letting $p_{L,r}^{(j)} = \P \big(X^{(j)}<q_{r}^{\star (j)}\big)$ and $p_{R,r}^{(j)} = \P \big(X^{(j)} \geq q_{r}^{\star (j)}\big)$ with $p_{L,r}^{(j)}, p_{R,r}^{(j)} >0$, we have 
\begin{align*}
\delta_{L} &= \frac{N_{n,L}}{a_n}(\overline{Y}_{L} - \mu)^2 \\
&= \frac{N_{n,L}}{a_n}(\overline{Y}_{L}^{\star} - \mu)^2 - 2\frac{N_{n,L}}{a_n} (\overline{Y}_{L}^{\star} - \overline{Y}_{L})(\overline{Y}_{L}^{\star} - \mu)
+ \frac{N_{n,L}}{a_n} (\overline{Y}_{L}^{\star} - \overline{Y}_{L})^2 \\
&= \frac{1}{p_{L,r}^{(j)}} \big( \frac{1}{a_n} \sum_{i=1}^{a_n} (Y_i - \mu) \mathds{1}_{X_i^{(j)} < q^{\star(j)}_{r}} \big)^2 + R_{L,r}^{(j)},
\end{align*}
where
\begin{align*}
R_{L,r}^{(j)} = & \big(\frac{a_n N_{n,L}}{N_{n,L}^{\star 2}} - \frac{1}{p_{n,L}}\big)
 \big( \frac{1}{a_n} \sum_{i=1}^{a_n} (Y_i - \mu) \mathds{1}_{X_i^{(j)} < q^{\star(j)}_{r}} \big)^2 \\
& - 2\frac{N_{n,L}}{a_n} (\overline{Y}_{L}^{\star} - \overline{Y}_{L})(\overline{Y}_{L}^{\star} - \mu)
+ \frac{N_{n,L}}{a_n} (\overline{Y}_{L}^{\star} - \overline{Y}_{L})^2
\end{align*}
By the law of large numbers, $\lim \limits_{n \to \infty} \frac{N_{n,L}^{\star}}{a_n} = p_{L,r}^{(j)}$ in probability.
Using Equation (\ref{proof_lemma2_eq:3}) in the proof of Lemma \ref{criterion_residual_1}, 
it comes that, in probability, $\lim \limits_{n \to \infty} \frac{N_{n,L}}{a_n} = p_{L,r}^{(j)} $, 
and consequently $\lim \limits_{n \to \infty} \frac{a_n N_{n,L}}{N_{n,L}^{\star 2}}   = \frac{1}{p_{L,r}^{(j)}}$.
Since $\sqrt{a_n} \frac{1}{a_n} \sum_{i=1}^{a_n} (Y_i - \mu) \mathds{1}_{X_i^{(j)} < q^{\star(j)}_{r}}$ converges in distribution to a normal distribution by the Central limit theorem, 
\begin{align*}
\lim \limits_{n \to \infty} a_n \big(\frac{a_n N_{n,L}}{N_{n,L}^{\star 2}} - \frac{1}{p_{L,r}^{(j)}}\big)
 \big( \frac{1}{a_n} \sum_{i=1}^{a_n} (Y_i - \mu) \mathds{1}_{X_i^{(j)} < q^{\star(j)}_{r}} \big)^2 = 0, \quad  \textrm{in probability.}    
\end{align*}
Furthermore, as for Equation (\ref{proof_lemma2_eq2_W}) in the proof of Lemma \ref{criterion_residual_1}, 
\begin{align*}
\sqrt{a_n} & |\overline{Y}_{L}^{\star} - \overline{Y}_{L}|
\\ &\leq \frac{a_n^2}{N_{n,L} N_{n,L}^{\star}} \underbrace{\frac{\sqrt{a_n}}{a_n^2} \sum_{i=1,l=1}^{a_n} Y_i 
\big|\mathds{1}_{X_i^{(j)} < q^{\star(j)}_{r}}\mathds{1}_{X_l^{(j)} < \hat{q}^{(j)}_{r}}
 - \mathds{1}_{X_i^{(j)} < \hat{q}^{(j)}_{r}}\mathds{1}_{X_l^{(j)} < q^{\star(j)}_{r}}\big|}_{\varepsilon_{L}},
\end{align*}
and 
\begin{align*}
\E[\varepsilon_{L}] \leq 2 \sqrt{a_n} \P\big(\hat{q}^{(j)}_{r} \leq X^{(j)} < q^{\star(j)}_{r}\big)
+ 2 \sqrt{a_n} \P\big(q^{\star(j)}_{r} \leq X^{(j)} < \hat{q}^{(j)}_{r}\big).
\end{align*}
According to Lemma \ref{convergence_quantile}, the right hand side term converges to $0$. Then, in probability,  $\lim \limits_{n \to \infty} \varepsilon_L = 0$.
Additionally, $\lim \limits_{n \to \infty} \frac{a_n^2}{N_{n,L} N_{n,L}^{\star}} = \frac{1}{p_{L,r}^{(j) 2}}$, and then, in probability,
\begin{align} \label{eq_R_term_2}
\lim \limits_{n \to \infty} \sqrt{a_n} (\overline{Y}_{L}^{\star} - \overline{Y}_{L}) = 0.
\end{align}
The second term of $a_n R_{L,r}^{(j)}$ writes
\begin{align*}
- a_n \times 2\frac{N_{n,L}}{a_n} (\overline{Y}_{L}^{\star} &- \overline{Y}_{L})(\overline{Y}_{L}^{\star} - \mu)
\\ &= - 2\frac{N_{n,L}}{a_n} \times \sqrt{a_n} (\overline{Y}_{L}^{\star} - \overline{Y}_{L}) \times \sqrt{a_n} (\overline{Y}_{L}^{\star} - \mu),
\end{align*}
where in probability, $\lim \limits_{n \to \infty} 2\frac{N_{n,L}}{a_n} = p_{L,r}^{(j)}$, $\lim \limits_{n \to \infty} \sqrt{a_n} (\overline{Y}_{L}^{\star} - \overline{Y}_{L}) = 0$ according to equation \ref{eq_R_term_2}, and $\sqrt{a_n} (\overline{Y}_{L}^{\star} - \mu)$ converges to a normal random variable from the central limit theorem.
 By Slutsky theorem, in probability,
$\lim \limits_{n \to \infty} - a_n \times 2\frac{N_{n,L}}{a_n} (\overline{Y}_{L}^{\star} - \overline{Y}_{L})(\overline{Y}_{L}^{\star} - \mu) = 0$.
Finally for the third term of $a_n R_{L,r}^{(j)}$ we also use equation \ref{eq_R_term_2} to conclude that in probability
\begin{align*}
\lim \limits_{n \to \infty} a_n \times \frac{N_{n,L}}{a_n} (\overline{Y}_{L}^{\star} - \overline{Y}_{L})^2 
= \lim \limits_{n \to \infty} \frac{N_{n,L}}{a_n} [\sqrt{a_n} (\overline{Y}_{L}^{\star} - \overline{Y}_{L})]^2 = 0
\end{align*}
Consequently,
\begin{align*}
\lim \limits_{n \to \infty} a_n R_{L,r}^{(j)} = 0.
\end{align*}
Symmetrically, we also have
\begin{align*}
\delta_{R} = \frac{1}{p_{R}} \big( \frac{1}{a_n} \sum_{i=1}^{a_n} (Y_i - \mu) \mathds{1}_{X_i^{(j)} \geq q^{\star(j)}_{r}} \big)^2 + R_{R,r}^{(j)},
\end{align*}
with $\lim \limits_{n \to \infty} a_n R_{R,r}^{(j)} = 0$, in probability.

Each component of $\textbf{L}_{n,\path_1}^{(\bestcut_1)}$ writes, with $(j,r) \in \bestcut_1 \setminus (j_1,r_1)$,
\begin{align*}
L&_{a_{n}}\big(\R^p,\hat{q}_{n,r}^{(j)}\big)  - L_{a_{n}}\big(\R^p,\hat{q}_{n,r_1}^{(j_1)}\big)
= \frac{1}{p_{L,r}^{(j)}} \big( \frac{1}{a_n} \sum_{i=1}^{a_n} (Y_i - \mu) \mathds{1}_{X_i^{(j)} < q^{\star(j)}_{r}} \big)^2 \\
&+ \frac{1}{p_{R,r}^{(j)}} \big( \frac{1}{a_n} \sum_{i=1}^{a_n} (Y_i - \mu) \mathds{1}_{X_i^{(j)} \geq q^{\star(j)}_{r}} \big)^2  
- \frac{1}{p_{L,r_1}^{(j_1)}} \big( \frac{1}{a_n} \sum_{i=1}^{a_n} (Y_i - \mu) \mathds{1}_{X_i^{(j_{1})} < q^{\star(j_{1})}_{r_{1}}} \big)^2 \\
&- \frac{1}{p_{R,r_1}^{(j_1)}} \big( \frac{1}{a_n} \sum_{i=1}^{a_n} (Y_i - \mu) \mathds{1}_{X_i^{(j_{1})} \geq q^{\star(j_{1})}_{r_{1}}} \big)^2 + R_{L,r}^{(j)} + R_{R,r}^{(j)} - R_{L,r_1}^{(j_1)} - R_{R,r_1}^{(j_1)}.
 \end{align*}

We explicitly write $\bestcut_1 = \{(j_k, r_k)\}_{k=1,...,c_1}$. Then $\textbf{L}_{n,\path_1}^{(\bestcut_1)}$ can be decomposed as
\begin{align*}
a_n \textbf{L}_{n,\path_1}^{(\bestcut_1)} = h_{\path_1}(\mathbf{V}_n) + \mathbf{R}_{n, \path_1},
\end{align*}
where for $k \in \{1,...,c_1\}$,
\begin{align*}
V_{n,2k-1} = \sqrt{\frac{a_n}{p_{L,r_k}^{(j_k)}}} \frac{1}{a_n} \sum_{i=1}^{a_n} (Y_i - \mu) \mathds{1}_{X_i^{(j_k)} < q^{\star(j_k)}_{r_k}}, \\
V_{n,2k} = \sqrt{\frac{a_n}{p_{R,r_k}^{(j_k)}}} \frac{1}{a_n} \sum_{i=1}^{a_n} (Y_i - \mu) \mathds{1}_{X_i^{(j_k)} \geq q^{\star(j_k)}_{r_k}}.
\end{align*}
$h_{\path_1}$ is a multivariate quadratic form defined as
\begin{align*}
h_{\path_1}: \left( \begin{array}{c} x_1 \\ \vdots \\ x_{2c_1} \end{array} \right) \rightarrow \left( \begin{array}{c}
x_{3}^2 + x_{4}^2 - x_1^2 - x_2^2\\ \vdots \\
x_{2k-1}^2 + x_{2k}^2 - x_{1}^2 - x_{2}^2 \\ \vdots \\
x_{2c_1-1}^2 + x_{2c_1}^2 - x_{1}^2 - x_{2}^2  \end{array}\right).
\end{align*}
and $R_{n, \path_1, k} = R_{L,r_k}^{(j_k)} + R_{R,r_k}^{(j_k)} - R_{L,r_1}^{(j_1)} - R_{R,r_1}^{(j_1)}$.

From the multivariate central limit theorem, $\mathbf{V}_n \stackrel[n\rightarrow\infty]{\mathscr{D}}{\longrightarrow} \mathbf{V}$,
where $\mathbf{V}$ is a gaussian vector of covariance matrix $\textrm{Cov}[\mathbf{Z}]$, and $\mathbf{Z}$ is defined as, for $k \in \{1,...,c_1\}$,
\begin{align*}
Z_{2k-1} = \frac{1}{\sqrt{p_{L,k}}} (Y - \E[Y]) \mathds{1}_{X^{(j_k)} < q_{r_k}^{\star (j_k)}}, 
Z_{2k} = \frac{1}{\sqrt{p_{R,k}}} (Y - \E[Y]) \mathds{1}_{X^{(j_k)} \geq q_{r_k}^{\star (j_k)}},
\end{align*}
with the simplified notations $p_{L,k} = p_{L,r_k}^{(j_k)}$ and $p_{R,k} = p_{R,r_k}^{(j_k)}$.

Finally, since $\lim \limits_{n \to \infty} \mathbf{R}_{n, \path_1} = \mathbf{0}$ in probability, from Slutsky's theorem and the continuous mapping theorem,
$a_{n} \textbf{L}_{n,\path_1}^{(\bestcut_1)} \stackrel[n\rightarrow\infty]{\mathscr{D}}{\longrightarrow} h_{\path_1}(\mathbf{V})$.
Note that, since $\bX$ has a strictly positive density, each component of $h_{\path_1}(\mathbf{V})$ has a strictly positive variance.
\end{proof}





\begin{proof}[Proof of Lemma  \ref{convergence_pn_theta_1}]
Consider a path $\path = (j_1, r_1, \cdot)$. Set $\theta^{(V)} = (\theta_1^{(V)},\cdot,\cdot) \in \Omega^{(V)}$, a realization of the randomization of the split direction. Recalling that the best split in a random tree is the one maximizing the CART-splitting criterion, condition on $\Theta^{(V)}=\theta^{(V)}$,
\begin{align}
\label{pn_theta_P1}
\{ \path_{1} \in T(\Theta,\mathscr{D}_{n}) \} 
=\underset{\substack{(j,r) \in \theta_{1}^{(V)} \times \{1, ..., q - 1\} \\ \qquad \quad \setminus{(j_1, r_1)}}}
{\bigcap}\big\{ L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)>L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big)\big\}.
\end{align}
We recall that, given $\theta^{(V)}$, we define the set of best theoretical cuts along the variables in $\theta^{(V)}_1$ as  
$$
\bestcut_{1}^{\star}\big(\theta_1^{(V)}\big) = \underset{(j,r) \in \theta_{1}^{(V)} \times \{1, ..., q - 1\}}{\textrm{argmax}} L^{\star}\big(\mathbb{R}^{p},q_{r}^{\star(j)}\big).
$$
Obviously if $(j_1,r_1) \notin \theta_{1}^{(V)} \times \{1, ..., q - 1\}$, the probability to select $\path_1$ in the empirical and theoretical tree is null.
In the sequel, we assume that $(j_1,r_1) \in \theta_{1}^{(V)} \times \{1, ..., q - 1\}$ and distinguish between four cases:
$(j_1,r_1)$ is not among the best theoretical cuts $\bestcut_{1}^{\star}\big(\theta_1^{(V)}\big)$, is the only element in $\bestcut_{1}^{\star}\big(\theta_1^{(V)}\big)$, is one element of $\bestcut_{1}^{\star}\big(\theta_1^{(V)}\big)$ with a positive value of the theoretical CART-splitting criterion, or finally, is one element of $\bestcut_{1}^{\star}\big(\theta_1^{(V)}\big)$ that all have a null value of the theoretical CART-splitting criterion.
\paragraph{Case 1} We assume that 
$(j_1, r_1) \notin \bestcut_{1}^{\star}\big(\theta_1^{(V)}\big)$. By definition of the theoretical random forest,
\begin{align}
\mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big) = 0    
\end{align}
Let $\big(j^{\star}, r^{\star} \big) \in \bestcut_{1}^{\star} \big(\theta_1^{(V)}\big)$, thus 
\begin{align*}
\varepsilon =L^{\star}\big(\mathds{R}^{p},q_{r^{\star}}^{\star(j^{\star})}\big) - L^{\star}\big(\mathds{R}^{p},q_{r_{1}}^{\star (j_{1})}\big) >0.
\end{align*}
Using equation (\ref{pn_theta_P1}), we have:
\begin{align*}
\mathbb{P}\big(\path_{1} \in T&\big(\Theta,\mathscr{D}_{n}\big)|\Theta^{(V)}=\theta^{(V)}\big)
\\
\leq \mathbb{P}\big(L_{a_{n}}&\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)>L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r^{\star}}^{(j^{\star})}\big)\big) \\
\leq \mathbb{P}\big(L_{a_{n}}&\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
- L^{\star}\big(\mathds{R}^{p},q_{r_{1}}^{\star (j_{1})}\big) - \epsilon
>L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r^{\star}}^{(j^{\star})}\big)
- L^{\star}\big(\mathds{R}^{p},q_{r^{\star}}^{\star(j^{\star})}\big)\big) \\
 \leq \mathbb{P}\big(  L_{a_{n}}&\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
- L^{\star}\big(\mathds{R}^{p},q_{r_{1}}^{\star (j_{1})}\big)    
 \\ & - \big( L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r^{\star}}^{(j^{\star})}\big)
- L^{\star}\big(\mathds{R}^{p},q_{r^{\star}}^{\star(j^{\star})}\big) \big) > \epsilon \big)
\end{align*}
Therefore, according to Lemma \ref{criterion_consistency_1},
\begin{align*}
\lim \limits_{n \to \infty} \mathbb{P}\big(\path_{1} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}=\theta^{(V)}\big) = 0
= \mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big)
\end{align*}
\paragraph{Case 2} We assume that 
$\bestcut_{1}^{\star}\big(\theta_1^{(V)}\big) = \big\{ (j_1, r_1) \big\}$. By definition of the theoretical random forest,  
\begin{align}
\mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big) = 1.    
\end{align}
Conditional on $\Theta^{(V)}=\theta^{(V)}$,
\begin{align*}
\left\lbrace\path_{1} \in T(\Theta,\mathscr{D}_{n})\right\rbrace^c =
\underset{\substack{(j,r) \in \theta_{1}^{(V)} \times \{1, ..., q - 1\} \\ \qquad \quad  \setminus{(j_1, r_1)}}}
{\bigcup}\big\{ L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
\leq L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big)\big\},   
\end{align*}
which leads to 
\begin{align}
1 - & \mathbb{P}\big(\path_{1} \in   T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}=\theta^{(V)}\big) \nonumber \\
& \leq \sum_{(j,r) \in \theta_{1}^{(V)} \times \{1, ..., q - 1\} \setminus{(j_1, r_1)}}
\mathbb{P}\big( L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
\leq L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big) \big). \label{proof_eq_decomp_prob}
\end{align}
From Lemma \ref{criterion_consistency_1}, for all $j\in\theta^{(V)}_0$, $r\in\{1, ..., q - 1\}$
such that $(j, r) \neq (j_1, r_1)$, in probability,
\begin{align}
\lim \limits_{n \to \infty} L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
- L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big) 
=  L^{\star}\big(\mathbb{R}^{p}, q_{r_{1}}^{\star(j_{1})}\big)
-  L^{\star}\big(\mathbb{R}^{p}, q_{r}^{\star(j)}\big) > 0. \label{proof_eq_comp_crit}
\end{align}
Using inequality (\ref{proof_eq_decomp_prob}) and equation (\ref{proof_eq_comp_crit}), we finally obtain,
\begin{align*}
\lim \limits_{n \to \infty}
\mathbb{P}\big(\path_{1} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}=\theta^{(V)}\big)
= 1 = \mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big).
\end{align*}
\paragraph{Case 3} We assume that 
$(j_1, r_1) \in \bestcut_{1}^{\star}\big(\theta_1^{(V)}\big)$, $\big| \bestcut_{1}^{\star}\big(\theta_1^{(V)}\big) \big| > 1$,
and $L^{\star}\big(\mathbb{R}^{p}, q_{r_{1}}^{\star(j_{1})}\big) > 0$.
On one hand, conditional on  $\Theta^{(V)}=\theta^{(V)}$,
\begin{align*}
\left\lbrace \path_{1} \in T(\Theta,\mathscr{D}_{n}) \right\rbrace
\subset \underset{(j, r) \in \bestcut_{1}^{\star}(\theta_1^{(V)}) \setminus{(j_1, r_1)}}
{\bigcap}\big\{ L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
>L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big)\big\}.
\end{align*}
On the other hand, conditional on $\Theta^{(V)}=\theta^{(V)}$,
\begin{align*}
\left\lbrace \path_{1} \in T(\Theta,\mathscr{D}_{n})\right\rbrace^c 
= \underset{(j, r) \in \bestcut_{1}^{\star}(\theta_1^{(V)}) \setminus{(j_1, r_1)}}
{\bigcup}\big\{ L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
\leq L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big)\big\}  \\
 \underset{(j, r) \in \theta_1^{(V)} \times \{1, ..., q - 1\} \setminus \bestcut_{1}^{\star}{(\theta_1^{(V)})}}
{\bigcup}\big\{ L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
\leq L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big)\big\}.
\end{align*}
Combining the two previous inclusions, 
\begin{align*}
0 & \leq \mathbb{P}\big(\underset{(j, r) \in \bestcut_{1}^{\star}(\theta_1^{(V)}) \setminus{(j_1, r_1)}}
{\bigcap}\big\{ L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
>L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big)\big\} \big)\\
& \quad - \mathbb{P}\big(\path_{1} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}=\theta^{(V)}\big)  \\ 
& \leq \sum_{(j, r) \in \theta_1^{(V)} \times \{1, ..., q - 1\} \setminus \bestcut_{1}^{\star}{(\theta_1^{(V)})}}
\mathbb{P}\big( L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
\leq L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big) \big).
\end{align*}
Using the same reasoning as in \textbf{Case 2}, we get
\begin{align*}
\lim \limits_{n \to \infty}  & \mathbb{P}\big(\underset{(j, r) \in \bestcut_{1}^{\star}(\theta_1^{(V)}) \setminus{(j_1, r_1)}}
{\bigcap}\big\{ L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
>L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big)\big\} \big) \\
& \quad - \mathbb{P}\big(\path_{1} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}
=\theta^{(V)}\big) = 0.
\end{align*}
We define the random vector $\textbf{L}_{n,\path_1}^{(\bestcut_1^{\star})}$ where each component is the difference between the empirical
 CART-splitting criterion for the splits $(j, r) \in \bestcut_1^{\star} \setminus (j_1, r_1)$ and $(j_1, r_1)$,
\begin{align*}
\textbf{L}_{n,\path_1}^{(\bestcut_1^{\star})} = \left(\begin{array}{c}
L_{a_{n}}\big(\mathbb{R}^{p},\hat{q}_{n,r}^{(j)}\big)-L_{a_{n}}\big(\mathbb{R}^{p},\hat{q}_{n,r_1}^{(j_1)}\big)
\end{array}\right)_{(j, r) \in \bestcut_1^{\star} \setminus (j_1, r_1)},
\end{align*}
then
\begin{align*}
\mathbb{P}\big(\underset{(j, r) \in \bestcut_{1}^{\star}(\theta_1^{(V)}) \setminus{(j_1, r_1)}}
{\bigcap}\big\{ L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
>L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big)\big\} \big) 
= \mathbb{P} \big( \textbf{L}_{n,\path_1}^{(\bestcut_1^{\star})} < \textbf{0} \big)
\end{align*}
From Lemma \ref{criterion_distribution_1} (case (a)),
\begin{align*}
\sqrt{a_{n}} \textbf{L}_{n,\path_1}^{(\bestcut_1^{\star})} \stackrel[n\rightarrow\infty]{\mathscr{D}}{\longrightarrow}\mathcal{N}\big(0, \Sigma\big).
\end{align*}
where for all $(j,r), (j',r') \in \bestcut_1^{\star} \setminus (j_1, r_1)$, each element of the covariance matrix $\Sigma$ is defined by
\begin{align*}
\Sigma_{(j,r), (j',r')} = \textrm{Cov}[Z_{j,r}, Z_{j',r'}],
\end{align*}
with
\begin{align*}
Z_{j,r}= & \big(Y - \mu_{L,r_1}^{(j_1)}\mathds{1}_{X^{(j_1)}<q_{r_1}^{\star(j_1)}}
-\mu_{R,r_1}^{(j_1)}\mathds{1}_{X^{(j_1)}\geq q_{r_1}^{\star(j_1)}}\big)^{2}\\
&- \big(Y - \mu_{L,r}^{(j)}\mathds{1}_{X^{(j)}<q_{r}^{\star(j)}}
-\mu_{R,r}^{(j)}\mathds{1}_{X^{(j)}\geq q_{r}^{\star(j)}}\big)^{2},
\end{align*}
$\mu_{L,r}^{(j)} = \E\big[Y | X^{(j)}<q_{r}^{\star(j)}\big]$, 
$\mu_{R,r}^{(j)} = \E\big[Y | X^{(j)} \geq q_{r}^{\star(j)}\big]$, 
and the variance of $Z_{j,r}$ is strictly positive.
If $\Phi_{\theta_1^{(V)}, (j_1, r_1) }$ is the c.d.f. of the multivariate normal distribution of covariance matrix $\Sigma$, we can conclude
\begin{align*}
\lim \limits_{n \to \infty} \mathbb{P}\big(\path_{1} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}
=\theta^{(V)}\big) & = \lim \limits_{n \to \infty} \mathbb{P} \big( \sqrt{a_{n}} \textbf{L}_{n,\path_1}^{(\bestcut_1^{\star})} < \textbf{0} \big)\\
& = \Phi_{\theta_1^{(V)}, (j_1, r_1)} (\textbf{0}),
\end{align*}
where  
\begin{align*}
\sum_{(j,r) \in \bestcut_{1}^{\star}(\theta_1^{(V)})} 
 \Phi_{\theta_1^{(V)}, (j, r)} (\textbf{0}) = 1.
\end{align*}
According to Definition \ref{def_proba_equality_1}, in the theoretical random forest, if $\bestcut_{1}^{\star}\big(\theta_1^{(V)}\big)$ has multiple elements, $(j_1, r_1)$ is randomly drawn with probability 
\begin{align*}
\mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big) 
= \Phi_{\theta_1^{(V)}, (j_1, r_1)} (\textbf{0}),
\end{align*}
that is 
\begin{align*}
\lim \limits_{n \to \infty} \mathbb{P}\big(\path_{1} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)} = \theta^{(V)}\big)
&= \mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big) 
\\ &= \Phi_{\theta_1^{(V)}, (j_1, r_1)} (\textbf{0}).
\end{align*}
We can notice that, in the specific case where $\bestcut_{1}^{\star}\big(\theta_1^{(V)}\big)$ has two elements, they are both selected with equal probability $\frac{1}{2}$. For more than two elements, the weights are not necessary equal, it depends on the covariance matrix $\Sigma$.
\paragraph{Case 4} We assume that all candidate splits have a null value for the theoretical CART-splitting criterion, 
i.e. for $(j, r) \in \theta_{1}^{(V)} \times \{1, ..., q - 1\}$, $L^{\star}\big(\mathbb{R}^{p}, q_{r}^{\star(j)}\big) = 0$.
Consequently $\bestcut_{1}^{\star}(\theta_1^{(V)}) = \theta_{1}^{(V)} \times \{1, ..., q - 1\}$.
By definition
\begin{align*}
\mathbb{P}\big(\path_{1} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}=\theta^{(V)}\big) 
= \P \big(\textbf{L}_{n,\path_1}^{(\bestcut_{1}^{\star})} < \mathbf{0}\big).
\end{align*}
According to Lemma \ref{criterion_distribution_1} (case (b)),
\begin{align*}
a_{n} \textbf{L}_{n,\path_1}^{(\bestcut_1)} \stackrel[n\rightarrow\infty]{\mathscr{D}}{\longrightarrow} h_{\path_1}(\mathbf{V}),
\end{align*}
where $\mathbf{V}$ is a gaussian vector of covariance matrix $\textrm{Cov}[\mathbf{Z}]$. If $\bestcut_{1}^{\star}\big(\theta_1^{(V)}\big)$ is explicitly written
$\bestcut_{1}^{\star}\big(\theta_1^{(V)}\big) = \{(j_k, r_k)\}_{k=1,...,c_1}$, $\mathbf{Z}$ is defined as, for $k \in \{1,...,c_1\}$,
\begin{align*}
Z_{2k-1} = \frac{1}{\sqrt{p_{L,k}}} (Y - \E[Y]) \mathds{1}_{X^{(j_k)} < q_{r_k}^{\star (j_k)}} \\
Z_{2k} = \frac{1}{\sqrt{p_{R,k}}} (Y - \E[Y]) \mathds{1}_{X^{(j_k)} \geq q_{r_k}^{\star (j_k)}},
\end{align*}
$p_{L,k} = \P \big(X^{(j_k)}<q_{r_k}^{\star (j_k)}\big)$, $p_{R,k} = \P \big(X^{(j_k)} \geq q_{r_k}^{\star (j_k)}\big)$, and $h_{\path_1}$ is a multivariate quadratic form defined as
\begin{align*}
h_{\path_1}: \left( \begin{array}{c} x_1 \\ \vdots \\ x_{2c_1} \end{array} \right) \rightarrow \left( \begin{array}{c}
x_{3}^2 + x_{4}^2 - x_1^2 - x_2^2\\ \vdots \\
x_{2k-1}^2 + x_{2k}^2 - x_{1}^2 - x_{2}^2 \\ \vdots \\
x_{2c_1-1}^2 + x_{2c_1}^2 - x_{1}^2 - x_{2}^2  \end{array}\right).
\end{align*}
and the variance of each component of $h_{\path_1}(\mathbf{V})$ is strictly positive.
If $\Phi_{\theta_1^{(V)}, (j_1, r_1)}$ is the cdf of $h_{\path_1}(\mathbf{V})$, then as in \textbf{Case 3},
\begin{align*}
\lim \limits_{n \to \infty} \mathbb{P}\big(\path_{1} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)} =\theta^{(V)}\big)
 &= \Phi_{\theta_1^{(V)}, (j_1, r_1)}(\mathbf{0})
\\ &=  \mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)} =\theta^{(V)}\big).
\end{align*}
\end{proof}




\subsubsection{Case 2: $\path_{2}$}



\begin{proof}[Proof of Lemma \ref{criterion_residual_2}] 

Let $(j,r) \in \bestcut_{\path_1}$.
\begin{align*}
& \sqrt{a_{n}}\big(L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big) - L_{a_{n}}\big(H^{\star}(\path_{1}),q_{r}^{\star(j)}\big)\big)\\
& =\sqrt{a_{n}}\big[L_{a_{n}}\big(H^{\star}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)-L_{a_{n}}\big(H^{\star}(\path_{1}),q_{r}^{\star(j)}\big)\big]\\
& \qquad +\sqrt{a_{n}}\big[L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)-L_{a_{n}}\big(H^{\star}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)\big].
\end{align*}
Since $(j,r) \in \bestcut_{\path_1}$, $\mathbb{P}\big(\bX \in H^{\star}(\path_{1}) | X^{(j)} < q_{r}^{\star(j)}\big)>0$
and $\mathbb{P}\big(\bX \in H^{\star}(\path_{1}) | X^{(j)} \geq q_{r}^{\star(j)}\big)>0$. Then, we can directly apply Lemma \ref{criterion_residual_1} to the first term of this decomposition, which shows that, in probability
\begin{align*}
 \lim\limits_{n \to \infty} \sqrt{a_{n}}\big(L_{a_{n}}\big(H^{\star}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)-L_{a_{n}}\big(H^{\star}(\path_{1}),q_{r}^{\star(j)}\big)\big) = 0.
\end{align*}

We expand the second term
\begin{align*}
& \sqrt{a_{n}}\big(L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big) - L_{a_{n}}\big(H^{\star}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)\big) = I_1 - I_2 - I_3 + I_4,
\end{align*}
where 
\begin{align*}
I_1 =\frac{\sqrt{a_{n}}}{N_{n}(\hat{H}_{n}(\path_{1}))}\sum_{i=1}^{a_{n}}\big(Y_{i}-\overline{Y}_{\hat{H}_{n}(\path_{1})}\big)^{2}\mathds{1}_{\bX_{i}\in\hat{H}_{n}(\path_{1})}
\end{align*}
\begin{align*}
I_2 = \frac{\sqrt{a_{n}}}{N_{n}(H^{\star}(\path_{1}))}\sum_{i=1}^{a_{n}}\big(Y_{i}-\overline{Y}_{H^{\star}(\path_{1})}\big)^{2}\mathds{1}_{\bX_{i}\in H^{\star}(\path_{1})}
\end{align*}
\begin{align*}
I_3 = \frac{\sqrt{a_{n}}}{N_{n}(\hat{H}_{n}(\path_{1}))}\sum_{i=1}^{a_{n}}\big(Y_{i}-\overline{Y}_{\hat{H}_{L}}\mathds{1}_{X_{i}^{(j)}<\hat{q}_{n,r}^{(j)}}-\overline{Y}_{\hat{H}_{R}}\mathds{1}_{X_{i}^{(j)}\geq\hat{q}_{n,r}^{(j)}}\big)^{2}\mathds{1}_{\bX_{i}\in\hat{H}_{n}(\path_{1})}
\end{align*}
\begin{align*}
I_4 = \frac{\sqrt{a_{n}}}{N_{n}(H^{\star}(\path_{1}))}\sum_{i=1}^{a_{n}}\big(Y_{i}-\overline{Y}_{H_{L}^{\star}}\mathds{1}_{X_{i}^{(j)}<\hat{q}_{n,r}^{(j)}}-\overline{Y}_{H_{R}^{\star}}\mathds{1}_{X_{i}^{(j)}\geq\hat{q}_{n,r}^{(j)}}\big)^{2}\mathds{1}_{\bX_{i}\in H^{\star}(\path_{1})}
\end{align*}
with $\hat{H}_{L}=\big\{ \bx\in\hat{H}_{n}(\path_{1}):x^{(j)}<\hat{q}_{n,r}^{(j)}\big\}$, $H_{L}^{\star}=\big\{ \bx\in H^{\star}(\path_{1}):x^{(j)}<\hat{q}_{n,r}^{(j)}\big\} $, and for all $H \subseteq \R^{p}$
\begin{align*}
N_{n}(H) = \frac{1}{a_{n}} \sum_{i=1}^{a_{n}}\mathds{1}_{\bX_{i} \in H}, \quad \overline{Y}_{H} = \frac{1}{N_{n}(H)} \sum_{i=1}^{a_{n}} Y_{i}\mathds{1}_{\bX_{i} \in H}.
\end{align*} 
We define symmetrically $\hat{H}_{R}$ and $H_{R}^{\star}$. We obtain
\begin{align*}
 \sqrt{a_{n}}&\big(L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big) - L_{a_{n}}\big(H^{\star}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)\big) \\
= & \underbrace{\sqrt{a_{n}}\big(\overline{Y}_{H^{\star}(\path_{1})}^{2}-\overline{Y}_{\hat{H}_{n}(\path_{1})}^{2}\big)}_{\Delta_{n,1}}\\
& + \underbrace{\sqrt{a_{n}}\frac{\overline{Y}_{\hat{H}_{L}}^{2}N_{n}(\hat{H}_{L})N_{n}(H^{\star}(\path_{1}))-\overline{Y}_{H_{L}^{\star}}^{2}N_{n}(H_{L}^{\star})N_{n}(\hat{H}_{n}(\path_{1}))}{N_{n}(\hat{H}_{n}(\path_{1}))N_{n}(H^{\star}(\path_{1}))}}_{\Delta_{n,2}}\\
& +\underbrace{\sqrt{a_{n}}\frac{\overline{Y}_{\hat{H}_{R}}^{2}N_{n}(\hat{H}_{R})N_{n}(H^{\star}(\path_{1}))-\overline{Y}_{H_{R}^{\star}}^{2}N_{n}(H_{R}^{\star})N_{n}(\hat{H}_{n}(\path_{1}))}{N_{n}(\hat{H}_{n}(\path_{1}))N_{n}(H^{\star}(\path_{1}))}}_{\Delta_{n,3}} 
\end{align*}
We first consider $\Delta_{n,1}$. Simple calculations show that 
\begin{align*}
\Delta_{n,1}  =& \frac{\sqrt{a_{n}}}{N_{n}(H^{\star}(\path_{1}))^{2}N_{n}(\hat{H}_{n}(\path_{1}))^{2}} \\
&\times \sum_{i,k,l,m} Y_{i}Y_{k}\big[\mathds{1}_{\bX_{i}\in H^{\star}(\path_{1}),\bX_{k}\in H^{\star}(\path_{1}),\bX_{l}\in\hat{H}_{n}(\path_{1}),\bX_{m}\in\hat{H}_{n}(\path_{1})}\\
& \qquad -\mathds{1}_{\bX_{i}\in\hat{H}_{n}(\path_{1}),\bX_{k}\in\hat{H}_{n}(\path_{1}),\bX_{l}\in H^{\star}(\path_{1}),\bX_{m}\in H^{\star}(\path_{1})}\big]
\end{align*}
We consider the case $s_{1}=L$, ($s_{1}=R$ is similar). Since  $Y_{i}\in\big\{ 0,1\big\}$,
\begin{align*}
|\Delta_{n,1}| \leq&\frac{\sqrt{a_{n}}}{N_{n}(H^{\star}(\path_{1}))^{2}N_{n}(\hat{H}_{n}(\path_{1}))^{2}} \\
& \times \sum_{i,k,l,m}\big|\mathds{1}_{X_{i}^{(j_{1})}<q_{r_{1}}^{\star(j_{1})},X^{(j_{1})}_{k}<q_{r_{1}}^{\star(j_{1})},X^{(j_{1})}_{l}<\hat{q}_{n,r_{1}}^{(j_{1})},X^{(j_{1})}_{m}<\hat{q}_{n,r_{1}}^{(j_{1})}}\\
& \qquad -\mathds{1}_{X^{(j_{1})}_{i}<\hat{q}_{n,r_{1}}^{(j_{1})},X^{(j_{1})}_{k}<\hat{q}_{n,r_{1}}^{(j_{1})},X^{(j_{1})}_{l}<q_{r_{1}}^{\star(j_{1})},X^{(j_{1})}_{m}<q_{r_{1}}^{\star(j_{1})}}\big|
\end{align*}

As in the proof of Lemma  \ref{criterion_residual_1}, according to Lemma \ref{convergence_quantile},  $\lim\limits_{n \to \infty} \Delta_{n,1} = 0$, in probability. Since $\Delta_{n,2}$ and $\Delta_{n,3}$ are the same quantities computed on each of the two daughter nodes, we study $\Delta_{n,2}$ only. 
\begin{align*}
\Delta_{n,2} & =\frac{\sqrt{a_{n}} (N_{n}(\hat{H}_{L})N_{n}(H_{L}^{\star}))^{-1}}{N_{n}(\hat{H}_{n}(\path_{1}))N_{n}(H^{\star}(\path_{1}))} \\
& \times \sum_{i,k,l,m}Y_{i}Y_{k}\big[\mathds{1}_{\bX_{i}\in\hat{H}_{L},\bX_{k}\in\hat{H}_{L},\bX_{l}\in H_{L}^{\star},\bX_{m}\in H^{\star}(\path_{1})}\\
& \qquad \quad -\mathds{1}_{\bX_{i}\in H_{L}^{\star},\bX_{k}\in H_{L}^{\star},\bX_{l}\in\hat{H}_{L},\bX_{m}\in\hat{H}_{n}(\path_{1})}\big] \\
& =\frac{\sqrt{a_{n}} (N_{n}(\hat{H}_{L})N_{n}(H_{L}^{\star}))^{-1}}{N_{n}(\hat{H}_{n}(\path_{1}))N_{n}(H^{\star}(\path_{1}))} 
\sum_{i,k,l,m}Y_{i}Y_{k}
\mathds{1}_{X^{(j)}_{i}<\hat{q}_{n,r}^{(j)},X^{(j)}_{k}<\hat{q}_{n,r}^{(j)},X^{(j)}_{l}<\hat{q}_{n,r}^{(j)}}\\
& \quad \times \big[\mathds{1}_{X^{(j_{1})}_{i}<\hat{q}_{n,r_{1}}^{(j_{1})},X^{(j_{1})}_{k}<\hat{q}_{n,r_{1}}^{(j_{1})},X^{(j_{1})}_{l}<q_{r_{1}}^{\star(j_{1})},X^{(j_{1})}_{m}<q_{r_{1}}^{\star(j_{1})}} \\
& \qquad -\mathds{1}_{X^{(j_{1})}_{i}<q_{r_{1}}^{\star(j_{1})},X^{(j_{1})}_{k}<q_{r_{1}}^{\star(j_{1})},X^{(j_{1})}_{l}<\hat{q}_{n,r_{1}}^{(j_{1})},X^{(j_{1})}_{m}<\hat{q}_{n,r_{1}}^{(j_{1})}}\big].
\end{align*}
Therefore
\begin{align*}
|\Delta_{n,2}| & \leq \frac{\sqrt{a_{n}} (N_{n}(\hat{H}_{L})N_{n}(H_{L}^{\star}))^{-1}}{N_{n}(\hat{H}_{n}(\path_{1}))N_{n}(H^{\star}(\path_{1}))} \\
& \times \sum_{i,k,l,m}\big|\mathds{1}_{X^{(j_{1})}_{i}<\hat{q}_{n,r_{1}}^{(j_{1})},X^{(j_{1})}_{k}<\hat{q}_{n,r_{1}}^{(j_{1})},X^{(j_{1})}_{l}<q_{r_{1}}^{\star(j_{1})},X^{(j_{1})}_{m}<q_{r_{1}}^{\star(j_{1})}}\\
& \qquad -\mathds{1}_{X^{(j_{1})}_{i}<q_{r_{1}}^{\star(j_{1})},X^{(j_{1})}_{k}<q_{r_{1}}^{\star(j_{1})},X^{(j_{1})}_{l}<\hat{q}_{n,r_{1}}^{(j_{1})},X^{(j_{1})}_{m}<\hat{q}_{n,r_{1}}^{(j_{1})}}\big|.
\end{align*}
As in the proof of Lemma \ref{criterion_residual_1}, according to Lemma \ref{convergence_quantile}, $\lim\limits_{n \to \infty} \Delta_{n,2} = 0$, in probability, which concludes the proof, since $\Delta_{n,3}$ can be studied in the same manner.  
\end{proof}







\begin{proof}[Proof of Lemma \ref{criterion_consistency_2}]
Let $(j,r) \in \bestcut_{\path_1}$.
\begin{align}
L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)
=&L_{a_{n}}\big(H^{\star}(\path_{1}),q_{r}^{\star(j)}\big)\nonumber \\
& +\big[L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)
-L_{a_{n}}\big(H^{\star}(\path_{1}),q_{r}^{\star(j)}\big)\big] \label{proof_eq_decomp_path2}
\end{align}
According to Lemma \ref{criterion_residual_2}, the second term in equation (\ref{proof_eq_decomp_path2}) converges to $0$ in probability. From the law of large numbers, in probability, 
\begin{align*}
\lim \limits_{n \to \infty} L_{a_{n}}\big(H^{\star}(\path_{1}),q_{r}^{\star(j)}\big)
=L^{\star}\big(H^{\star}(\path_{1}),q_{r}^{\star(j)}\big),
\end{align*}
which concludes the proof. 
\end{proof}


\begin{proof}[Proof of Lemma \ref{criterion_distribution_2}]
Similar to the case with $\path_1$ (Lemma \ref{criterion_consistency_1}), where Lemma \ref{criterion_residual_2} is used instead of Lemma \ref{criterion_residual_1}.
\end{proof}


\begin{proof}[Proof of Lemma \ref{convergence_pn_theta_2}]

Consider a path $\path_2 = \{(j_1, r_1, L), (j_2, r_2, \cdot)\}$. Set $\theta^{(V)} = \big(\theta_1^{(V)}, \theta_2^{(V)}\big)$, a realization of  $\Theta^{(V)} = \big(\Theta_0^{(V)}, \Theta_L^{(V)}\big)$, the randomization of the split direction at the root node and its left child node--- see Section 2,
and note that we omit $\Theta_R^{(V)}$ to simplify notations. Then, $\theta_1^{(V)}$ and $\theta_2^{(V)}$  denotes the set of eligible variables for 
respectively the first and second split. We also consider $\bestcut_{\path_1}\big(\theta_2^{(V)}\big) \subset \bestcut_{\path_1}$ the set of eligible second splits.

Recalling that the best split in a random tree is the one maximizing the CART-splitting criterion, conditional on $\Theta^{(V)}=\theta^{(V)}$,
\begin{align*}
\left\lbrace\path_{2} \in T(\Theta,\mathscr{D}_{n})\right\rbrace 
 = \underset{\substack{(j, r) \in \theta_{1}^{(V)}  \times \{1, ..., q - 1\} \\ \qquad \quad \setminus (j_1, r_1)}}
{\bigcap}\big\{ L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)>L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big)\big\} \\
 \qquad \underset{(j, r) \in \bestcut_{\path_1}(\theta_2^{(V)}) \setminus (j_2, r_2)}
{\bigcap}\big\{ L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r_{2}}^{(j_{2})}\big)
>L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)\big\}
\end{align*}
Recall that $\bestcut_{1}^{\star}\big(\theta_{1}^{(V)}\big) = \underset{ (j,r) \in\theta_{1}^{(V)} \times \{1, ..., q - 1\}}{\textrm{argmax}} L^{\star}\big(\mathbb{R}^{p},q_{r}^{\star(j)}\big) $, and similarly
\begin{align*}
\bestcut_{2}^{\star}\big(\theta_{2}^{(V)}\big) = \underset{ (j,r) \in \bestcut_{\path_1}(\theta_2^{(V)})}{\textrm{argmax}}
 L^{\star}\big(H^{\star}(\path_{1}),q_{r}^{\star(j)}\big).    
\end{align*}

Obviously if $(j_1,r_1) \notin \theta_{1}^{(V)} \times \{1, ..., q - 1\}$ or 
$(j_2,r_2) \notin \bestcut_{\path_1}(\theta_2^{(V)}) $, the probability to select $\path_2$ in the empirical and theoretical tree is null.
In the sequel, we assume that $(j_1,r_1) \in \theta_{0}^{(V)} \times \{1, ..., q - 1\}$ and 
$(j_2,r_2) \in \bestcut_{\path_1}\big(\theta_2^{(V)}\big)$ and distinguish between cases,
 depending on whether $(j_1, r_1) \in \bestcut_{1}^{\star}\big(\theta_{1}^{(V)}\big)$
 or not and $(j_2, r_2) \in \bestcut_2^{\star}\big(\theta_2^{(V)}\big)$ or not, as well as the cardinality of $\bestcut_1^{\star}\big(\theta_1^{(V)}\big)$ and
$\bestcut_2^{\star}\big(\theta_2^{(V)}\big)$, and whether the maximum of the theoretical CART-splitting criterion is null or not.

\paragraph{Case 1} We assume that $(j_1, r_1) \notin \bestcut_{1}^{\star}\big(\theta_{1}^{(V)}\big)$. Hence, the theoretical decision tree satisfies 
$$\mathbb{P}\big(\path_{2} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big) = \mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big) = 0.$$
According to Lemma \ref{convergence_pn_theta_1}, we have
\begin{align*}
\lim \limits_{n \to \infty} \mathbb{P}&\big(\path_{2} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}=\theta^{(V)}\big)\\
& \leq \lim \limits_{n \to \infty} \mathbb{P}\big(\path_{1} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}=\theta^{(V)}\big)\\
& = 0\\
& = \mathbb{P}\big(\path_{2} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big).    
\end{align*}


\paragraph{Case 2} We assume that $\big(j_{2}, r_{2}\big) \notin \bestcut_{2}^{\star}\big(\theta_{2}^{(V)}\big)$. Again, for the theoretical decision tree, $$\mathbb{P}\big(\path_{2} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big) = 0.$$
Letting $\big(j^{\star}, r^{\star} \big) \in \bestcut_{2}^{\star}\big(\theta_{2}^{(V)}\big)$, 
\begin{align*}
\varepsilon
=L^{\star}\big(H^{\star}(\path_{1}),q_{r^{\star}}^{\star(j^{\star})}\big) - L^{\star}\big(H^{\star}(\path_{1}),q_{r_{2}}^{\star (j_{2})}\big).
\end{align*}
Therefore, 
\begin{align*}
 \mathbb{P}\big(\path_{2}& \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}=\theta^{(V)}\big) \\
& \leq \mathbb{P}\big(L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r_{2}}^{(j_{2})}\big)
>L_{a_{n}}\big(H^{\star}(\path_{1}),\hat{q}_{n,r^{\star}}^{(j^{\star})}\big)\big) \\
& \leq \mathbb{P}\big(L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r_{2}}^{(j_{2})}\big)
- L^{\star}\big(H^{\star}(\path_{1}),q_{r_{2}}^{\star (j_{2})}\big) - \epsilon  \\ & \qquad 
>L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r^{\star}}^{(j^{\star})}\big)
- L^{\star}\big(H^{\star}(\path_{1}),q_{r^{\star}}^{\star(j^{\star})}\big)\big) \\
& \leq \mathbb{P}\big( L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r_{2}}^{(j_{2})}\big)
- L^{\star}\big(H^{\star}(\path_{1}),q_{r_{2}}^{\star (j_{2})}\big)  \\ 
& \qquad -  \big( L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r^{\star}}^{(j^{\star})}\big)
- L^{\star}\big(H^{\star}(\path_{1}),q_{r^{\star}}^{\star(j^{\star})}\big) \big)
> \epsilon \big).
\end{align*}
Consequently, according to Lemma \ref{criterion_consistency_2}, 
\begin{align*}
\lim \limits_{n \to \infty} \mathbb{P}\big(\path_{2} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}=\theta^{(V)}\big) = 0
= \mathbb{P}\big(\path_{2} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big).
\end{align*}


\paragraph{Case 3} We assume that
$(j_1, r_1) \in \bestcut_{1}^{\star}\big(\theta_{1}^{(V)}\big)$
and $\bestcut_{2}^{\star}\big(\theta_{2}^{(V)}\big) = \{ (j_{2}, r_{2}) \}$,
i.e. $(j_{2}, r_{2})$ is the unique maximum of the theoretical CART-splitting criterion for the cell $H^{\star}(\path_{1})$. By definition of the theoretical decision tree,
\begin{align*}  
\mathbb{P}\big(\path_{2} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big)
= \mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big)
\end{align*}
Conditional on $\{\Theta^{(V)}=\theta^{(V)}\}$, 
\begin{align}
\{ \path_{2}& \in T(\Theta,\mathscr{D}_{n}) \} 
=  \big\{ \path_{1} \in T(\Theta,\mathscr{D}_{n}) \big\} \nonumber \\
& \qquad 
\underset{(j, r) \in \bestcut_{\path_1}(\theta_2^{(V)})  \setminus (j_2, r_2)}
{\bigcap}\big\{ L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r_{2}}^{(j_{2})}\big)
>L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)\big\}. \label{proof_twosplits_prob_ineq3}
\end{align}
Consequently, 
\begin{align}
& \mathbb{P}\big(\path_{2} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}=\theta^{(V)}\big) \nonumber \\
& \geq \mathbb{P} \big( \path_{1} \in T(\Theta,\mathscr{D}_{n})\big |\Theta^{(V)}=\theta^{(V)} \big) \nonumber \\
& - \sum_{(j, r) \in \bestcut_{\path_1}(\theta_2^{(V)})  \setminus (j_2, r_2)}
\mathbb{P}\big( L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r_{2}}^{(j_{2})}\big)
\leq L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big) \big). \label{proof_twosplits_prob_ineq}
\end{align}
For $(j, r) \in \bestcut_{\path_1}\big(\theta_2^{(V)}\big)  \setminus (j_2, r_2)$,
\begin{align}
L^{\star}\big(H^{\star}(\path_{1}),q_{r_{2}}^{\star(j_{2})}\big)
-  L^{\star}\big(H^{\star}(\path_{1}),q_{r}^{\star(j)}\big)> 0. \label{proof_twosplits_prob_ineq2}
\end{align}
Thus, using inequalities (\ref{proof_twosplits_prob_ineq}) and (\ref{proof_twosplits_prob_ineq2}), and according to 
Lemma \ref{criterion_consistency_2}, 
\begin{align*}
\lim \limits_{n \to \infty} \mathbb{P}\big( L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r_{2}}^{(j_{2})}\big) 
\leq L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big) \big) = 0,
\end{align*}
and thus, using (\ref{proof_twosplits_prob_ineq3}) and (\ref{proof_twosplits_prob_ineq}), 
\begin{align*}
\lim \limits_{n \to \infty}& \mathbb{P}\big(\path_{2} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}=\theta^{(V)}\big) \\
& = \lim \limits_{n \to \infty} \mathbb{P} \big( \path_{1} \in T(\Theta,\mathscr{D}_{n}) |\Theta^{(V)}=\theta^{(V)} \big)
 = \mathbb{P} \big( \path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)} \big) \\
& = \mathbb{P}\big(\path_{2} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big),
\end{align*}
where the second inequality is a direct consequence of Lemma \ref{convergence_pn_theta_1}.


\paragraph{Case 4} For the first split, we assume
$(j_1, r_1) \in \bestcut_{1}^{\star}\big(\theta_{1}^{(V)}\big)$ with $L^{\star}\big(\R^p, q^{\star(j_1)}_{r_1}\big) > 0$, and for the second split,
$\big(j_{2}, r_{2}\big) \in \bestcut_{2}^{\star}\big(\theta_{2}^{(V)}\big)$ with $| \bestcut_{2}^{\star}\big(\theta_{2}^{(V)}\big) \big| > 1$
and $L^{\star}\big(H^{\star}(\path_1), q^{\star(j_2)}_{r_2}\big) > 0$.

On one hand, conditional on the event $\{ \Theta^{(V)} = \theta^{(V)} \}$,
\begin{align}
\left\lbrace \path_{2} \in T(\Theta,\mathscr{D}_{n}) \right\rbrace =
 \underset{\substack{(j, r) \in \theta_1^{(V)} \times \{1, ..., q - 1\} \\ \qquad \quad\setminus (j_1, r_1)}}
 {\bigcap}\big\{ L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
>L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big)\big\} \nonumber \\
 \qquad \underset{(j, r) \in \bestcut_{\path_1}(\theta_2^{(V)}) \setminus (j_2, r_2)}
{\bigcap}\big\{ L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r_{2}}^{(j_{2})}\big)
>L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)\big\}. \label{proof_thm_twosplits_case4_eq1}
\end{align}
Using equation (\ref{proof_thm_twosplits_case4_eq1}) to find a subset and a superset of $\left\lbrace \path_{2} \in T(\Theta,\mathscr{D}_{n}) \right\rbrace$, we obtain
\begin{align*}
0 & \geq  \mathbb{P}\big(\path_{2} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}=\theta^{(V)}\big) \\
& \enskip  - \mathbb{P}\Bigg(\underset{(j, r) \in \bestcut_{1}^{\star}(\theta_{1}^{(V)}) \setminus (j_1, r_1)}
{\bigcap}\big\{ L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
>L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big)\big\}  \\ 
& \qquad \underset{(j, r) \in \bestcut_{2}^{\star}(\theta_{2}^{(V)}) \setminus (j_2, r_2)}
{\bigcap}\big\{ L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r_{2}}^{(j_{2})}\big)
>L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)\big\} \Bigg)   \\
& \geq \sum_{(j,r) \in\theta_{1}^{(V)} \times \{1, ..., q - 1\} \setminus \bestcut_{1}^{\star}{(\theta_{1}^{(V)})}}
\mathbb{P}\big( L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
\leq L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big) \big) \\
& \enskip + \sum_{(j,r) \in\theta_{2}^{(V)} \times \{1, ..., q - 1\} \setminus \bestcut_{2}^{\star}{(\theta_{2}^{(V)})}}
\mathbb{P}\big( L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r_{2}}^{(j_{2})}\big)
\leq L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big) \big)
\end{align*}
We  proved in \textbf{Case 3} that the limit of the last two terms of the previous inequality is zero, in probability. Therefore, 
\begin{align}
& \lim \limits_{n \to \infty} \mathbb{P}\big(\path_{2} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}=\theta^{(V)}\big) \nonumber \\
 & = \lim \limits_{n \to \infty} \mathbb{P}\Bigg(\underset{(j, r) \in \bestcut_{1}^{\star}(\theta_{1}^{(V)}) \setminus (j_1, r_1)}
 {\bigcap}\big\{ L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
>L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big)\big\}  \nonumber \\ 
& \qquad \underset{(j, r) \in \bestcut_{2}^{\star}(\theta_{2}^{(V)}) \setminus (j_2, r_2)}
{\bigcap}\big\{ L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r_{2}}^{(j_{2})}\big)
>L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)\big\} \Bigg). \label{proof_twosplits_eq2}
\end{align}

We define the random vector $\textbf{L}_{n,\path_2}^{(\bestcut_1^{\star}, \bestcut_2^{\star})}$ 
(we drop $\theta^{(V)}$ to lighten notations)
where each component is the difference between the empirical CART-splitting criterion for the splits $(j, r) \in \bestcut_1^{\star} \setminus (j_1, r_1)$ and $(j_1, r_1)$
 for the first $|\bestcut_1^{\star}| - 1$ components, and for the splits $(j, r) \in \bestcut_2^{\star} \setminus (j_2, r_2)$ and $(j_2, r_2)$ for the remaining $|\bestcut_2^{\star}| - 1$ components,  i.e.,
\begin{align*}
\textbf{L}_{n,\path_2}^{(\bestcut_1^{\star}, \bestcut_2^{\star})} = \left(\begin{array}{c}
\big[L_{a_{n}}\big(\mathbb{R}^{p},\hat{q}_{n,r}^{(j)}\big)-L_{a_{n}}\big(\mathbb{R}^{p},\hat{q}_{n,r_1}^{(j_1)}\big)\big]_{(j, r) \in \bestcut_1^{\star} \setminus (j_1, r_1)} \\
\big[L_{a_{n}}\big(\hat{H}_n(\path_{1}),\hat{q}_{n,r}^{(j)}\big)-L_{a_{n}}\big(\hat{H}_n(\path_{1}),\hat{q}_{n,r_2}^{(j_2)}\big)\big]_{(j, r) \in \bestcut_2^{\star} \setminus (j_2, r_2)} \\
\end{array}\right).
\end{align*}

Then, we can write
\begin{align}
& \mathbb{P}\Bigg(\underset{(j, r) \in \bestcut_{1}^{\star}(\theta_{1}^{(V)}) \setminus (j_1, r_1)}
{\bigcap}\big\{ L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r_{1}}^{(j_{1})}\big)
>L_{a_{n}}\big(\mathds{R}^{p},\hat{q}_{n,r}^{(j)}\big)\big\} \nonumber \\ 
& \qquad 
\underset{(j, r) \in \bestcut_{2}^{\star}(\theta_{2}^{(V)}) \setminus (j_2, r_2)}{\bigcap}\big\{ L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r_{2}}^{(j_{2})}\big)
>L_{a_{n}}\big(\hat{H}_{n}(\path_{1}),\hat{q}_{n,r}^{(j)}\big)\big\} \Bigg) \nonumber\\
&= \mathbb{P} \big( \textbf{L}_{n,2}^{(\bestcut_{1}^{\star}, \bestcut_{2}^{\star})} < \textbf{0} \big) \label{proof_twosplits_eq3}
\end{align}

According to Lemma \ref{criterion_distribution_2},
\begin{align*}
\sqrt{a_{n}} \textbf{L}_{n,\path_2}^{(\bestcut_1^{\star}, \bestcut_2^{\star})} \stackrel[n\rightarrow\infty]{\mathscr{D}}{\longrightarrow}\mathcal{N}(0, \Sigma)
\end{align*}
where for $l, l' \in \{1,2\}$, for all $(j,r) \in \bestcut_l^{\star} \setminus (j_l,r_l)$,
$(j', r') \in \bestcut_{l'}^{\star} \setminus (j_{l'}, r_{l'})$, each element of the covariance matrix $\Sigma$ is defined by
$ \Sigma_{(j,r,l), (j',r',l')} = \textrm{Cov}[Z_{j,r,l}, Z_{j',r',l'}],$ with
\begin{align*}
Z_{j,r,l}= & \frac{1}{\P(\bX \in H_l)} \big(Y - \mu_{L,r_l}^{(j_l)}\mathds{1}_{X^{(j_{l})}<q_{r_{l}}^{\star(j_{l})}}
-\mu_{R,r_l}^{(j_l)}\mathds{1}_{X^{(j_{l})}\geq q_{r_{l}}^{\star(j_{l})}}\big)^{2} \mathds{1}_{\bX \in H_l}  \\
&- \frac{1}{\P(\bX \in H_l)} \big(Y - \mu_{L,r}^{(j)}\mathds{1}_{X^{(j)}<q_{r}^{\star(j)}}
-\mu_{R,r}^{(j)}\mathds{1}_{X^{(j)}\geq q_{r}^{\star(j)}}\big)^{2} \mathds{1}_{\bX \in H_l},
\end{align*}
$\mu_{L,r}^{(j)} = \E\big[Y | X^{(j)}<q_{r}^{\star(j)}, \bX \in H_l\big]$, 
$\mu_{R,r}^{(j)} = \E\big[Y | X^{(j)} \geq q_{r}^{\star(j)}, \bX \in H_l\big]$, and the variance of $Z_{j,r,l}$ is strictly positive.

Letting $\Phi_{\path_{1}, \theta^{(V)}, (j_{2}, r_{2})}$ be the c.d.f. of the multivariate normal distribution with covariance matrix $\Sigma$,
and using equalities (\ref{proof_twosplits_eq2}) and (\ref{proof_twosplits_eq3}),
\begin{align*}
\lim \limits_{n \to \infty} \mathbb{P}\big(\path_{2} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}
=\theta^{(V)}\big) = \Phi_{\path_{1}, \theta^{(V)}, (j_{2}, r_{2})} (\textbf{0}).
\end{align*}
We can check that 
\begin{align*}
\sum_{(j,r) \in \bestcut_2^{\star}(\theta^{(V)})} 
 \Phi_{\path_{1}, \theta^{(V)}, (j, r)} (\textbf{0})
=\mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big). 
\end{align*}
In the theoretical random forest, the first cut $(j_1, r_1)$ is randomly selected with probability
$\mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big)$ (see the proof of Lemma \ref{convergence_pn_theta_1}).
For the second cut, according to Definition \ref{def_proba_equality_2}, if $\bestcut_{2}^{\star}\big(\theta_{2}^{(V)}\big)$ has multiple elements,
$(j_{2}, r_{2})$ is randomly drawn with probability 
\begin{align*}
\frac{\Phi_{\path_{1}, \theta^{(V)}, (j_{2}, r_{2})} (\textbf{0})}
{\mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big) }
\end{align*}
Since the random selection at the root node of the tree and its children nodes are independent in the theoretical algorithm,
$\path_{2}$ is selected with probability
\begin{align*}
\mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big)
\times \frac{\Phi_{\path_{1}, \theta^{(V)}, (j_{2}, r_{2})} (\textbf{0})}
{\mathbb{P}\big(\path_{1} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big) } \\
= \Phi_{\path_{1}, \theta^{(V)}, (j_{2}, r_{2})} (\textbf{0}).
\end{align*}
Ultimately, 
\begin{align*}
\lim \limits_{n \to \infty} \mathbb{P}\big(\path_{2} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)} = \theta^{(V)}\big)
& = \mathbb{P}\big(\path_{2} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big) \\
& = \Phi_{\path_{1}, \theta^{(V)}, (j_{2}, r_{2})} \big(\textbf{0}\big).
\end{align*}


\paragraph{Case 5} We assume that $(j_1, r_1) \in \bestcut_{1}^{\star}\big(\theta_{1}^{(V)}\big)$ and
$(j_{2}, r_{2}) \in \bestcut_{2}^{\star}\big(\theta_{2}^{(V)}\big)$, and that the theoretical CART-splitting criterion is null for both splits:
$L^{\star}\big(\R^p, q^{\star(j_1)}_{r_1}\big) = 0$ and  $L^{\star}\big(H^{\star}(\path_1), q^{\star(j_2)}_{r_2}\big) = 0$. 

Consequently $\bestcut_{1}^{\star}\big(\theta_1^{(V)}\big) = \theta_{1}^{(V)} \times \{1, ..., q - 1\}$,
and $\bestcut_{2}^{\star}\big(\theta_2^{(V)}\big) = \bestcut_{\path_1}\big(\theta_2^{(V)}\big)$.
Using the same notations defined in \textbf{Case 4}, we have by definition
\begin{align*}
\mathbb{P}\big(\path_{1} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)}=\theta^{(V)}\big) 
= \P \big(\textbf{L}_{n,\path_2}^{(\bestcut_{1}^{\star}, \bestcut_{2}^{\star})} < \mathbf{0}\big).
\end{align*}
According to Lemma \ref{criterion_distribution_2} (case (b)),
\begin{align*}
a_{n} \textbf{L}_{n,\path_2}^{(\bestcut_1^{\star}, \bestcut_2^{\star})} \stackrel[n\rightarrow\infty]{\mathscr{D}}{\longrightarrow} h_{\path_2}(\mathbf{V}),
\end{align*}
where $\mathbf{V}$ is a gaussian vector of covariance matrix $\textrm{Cov}[\mathbf{Z}]$. If $\bestcut_1^{\star}$ and $\bestcut_2^{\star}$ are explicitly written
$\bestcut_1^{\star} = \{(j_k, r_k)\}_{k \in J_1}$, and $\bestcut_2^{\star} = \{(j_k, r_k)\}_{k \in J_2}$, with $J_1 = \{1,...,c_1+1\} \setminus 2$
and $J_2 = \{2\} \cup \{c_1+2,...,c_1+c_2\}$, $\mathbf{Z}$ is defined as, for $l \in \{1,2\}$ and $k \in J_l$,
\begin{align*}
Z_{2k-1} = \frac{1}{\sqrt{p_{L,k}\P(\bX \in H_l)}} (Y - \E[Y|\bX \in H_l]) \mathds{1}_{X^{(j_k)} < q_{r_k}^{\star (j_k)}}\mathds{1}_{\bX \in H_l}, \\
Z_{2k} = \frac{1}{\sqrt{p_{R,k}\P(\bX \in H_l)}} (Y - \E[Y|\bX \in H_l]) \mathds{1}_{X^{(j_k)} \geq q_{r_k}^{\star (j_k)}}\mathds{1}_{\bX \in H_l},
\end{align*}
$p_{L,k} = \P \big(X^{(j_k)}<q_{r_k}^{\star (j_k)}, \bX \in H_l\big)$, $p_{R,k} = \P\big (X^{(j_k)} \geq q_{r_k}^{\star (j_k)}, \bX \in H_l\big)$.
$h_{\path_2}$ is a multivariate quadratic form defined as
\begin{align*}
h_{\path_2}: \left( \begin{array}{c} x_1 \\ \vdots \\ x_{2(c_1 + c_2)}  \end{array} \right) \rightarrow \left( \begin{array}{c}
x_{5}^2 + x_{6}^2 - x_1^2 - x_2^2\\ \vdots \\
x_{2c_1+1}^2 + x_{2c_1+2}^2 - x_{1}^2 - x_{2}^2\\ 
x_{2c_1+3}^2 + x_{2c_1+4}^2 - x_{3}^2 - x_{4}^2\\ \vdots \\
x_{2(c_1+c_2)-1}^2 + x_{2(c_1+c_2)}^2 - x_{3}^2 - x_{4}^2  \end{array}\right),
\end{align*}
and the variance of each component of $h_{\path_2}(\mathbf{V})$ is strictly positive.

$\Phi_{\path_{1}, \theta^{(V)}, (j_{2}, r_{2})}$ is now defined as the cdf of $h_{\path_2}(\mathbf{V})$,
and the end of the proof is identical to \textbf{Case 4}. We conclude
\begin{align*}
\lim \limits_{n \to \infty} \mathbb{P}\big(\path_{2} \in T(\Theta,\mathscr{D}_{n})|\Theta^{(V)} = \theta^{(V)}\big)
& = \mathbb{P}\big(\path_{2} \in T^{\star}(\Theta)|\Theta^{(V)}=\theta^{(V)}\big) \\
& = \Phi_{\path_{1}, \theta^{(V)}, (j_{2}, r_{2})} (\textbf{0}).
\end{align*}


\paragraph{Case 6} We assume $(j_1, r_1) \in \bestcut_{1}^{\star}\big(\theta_{1}^{(V)}\big)$, $(j_2, r_2) \in \bestcut_{2}^{\star}\big(\theta_{2}^{(V)}\big)$ and $\big|\bestcut_{2}^{\star}\big(\theta_{2}^{(V)}\big)\big| > 1$ as in \textbf{Case 4}, but either 
 $L^{\star}\big(\R^p, q^{\star(j_1)}_{r_1}\big) = 0$ and $L^{\star}\big(H^{\star}(\path_1), q^{\star(j_2)}_{r_2}\big) > 0$, or
 $L^{\star}\big(\R^p, q^{\star(j_1)}_{r_1}\big) > 0$ and $L^{\star}\big(H^{\star}(\path_1), q^{\star(j_2)}_{r_2}\big) = 0$.

The same reasoning than for \textbf{Cases 4} and \textbf{5} applies where the limit law of
$\textbf{L}_{n,\path_2}^{(\bestcut_1^{\star}, \bestcut_{2}^{\star})}$ has both gaussian and $\chi$-square components and is given by 
case (c) or case (d) of Lemma \ref{criterion_distribution_2}. 


\end{proof}



\section{Proof of Theorem \ref{stability_equivalent}}
\label{sec_stability_equivalent}

We recall Theorem \ref{stability_equivalent} for the sake of clarity.
\begin{theorem}
\label{stability_equivalent}
If $p_{0}\in[0, 1] \setminus \setpemp$ and $\mathscr{D}'_{n} = \mathscr{D}_{n}$, then, conditional on $\mathscr{D}_{n}$, we have
\begin{align} \label{eq_stab_M}
\lim \limits_{M \to \infty} \hat{S}_{M,n,p_{0}} = 1 \quad \textrm{in probability.}
\end{align}
In addition for $p_{0} < \max \setpemp$,
\begin{align*}
1 - & \mathbb{E} [ \hat{S}_{M,n,p_{0}} | \mathscr{D}_{n}] \\ & \underset{M \to \infty}{\sim} \sum_{\path \in \Pi}
\frac{\Phi( M p_{0}, M, p_n(\path)) ( 1 - \Phi\big( M p_{0}, M,
p_n(\path)))}
{\frac{1}{2} \sum_{\path' \in \Pi} \mathds{1}_{p_n(\path') > p_{0}} +
\mathds{1}_{p_n(\path') > p_{0} - \rho_{n}(\path, \path')
\frac{\sigma_{n}( \path')}{\sigma_{n}( \path) }(p_{0} - p_n(\path)) }},
\end{align*}
where $\Phi ( M p_{0}, M, p_n(\path))$ is the cdf of a binomial distribution with parameter 
$p_n(\path)$, $M$ trials, evaluated at $M p_{0}$, and, for all $\path, \path' \in \Pi$, 
\[
\sigma_{n}( \path) = \sqrt{p_n(\path) (1 -  p_n(\path))},
\]
and
\[
 \rho_{n}(\path, \path') = \frac{\emph{Cov}(\mathds{1}_{\path \in T(\Theta, \mathscr{D}_{n})}, 
\mathds{1}_{\path' \in T(\Theta, \mathscr{D}_{n})} | \mathscr{D}_{n} )}
{\sigma_{n}( \path ) \sigma_{n}( \path')}.
\]
\end{theorem}

Let  $p_{0}\in[0, \max ~\setpemp)
\setminus \setpemp$ and $\Dn' = \Dn$. 
Before proving Theorem \ref{stability_equivalent}, we need the following two lemmas. 

\begin{lemme} \label{F_limit}
Let $F$ be the hypergeometric function. Then, for $(a, c) \in \mathbb{Z}^2$ and $\path \in \Pi$ such that $p_n(\path) > p_0$, we have
\begin{align*}
\lim \limits_{M \to \infty} \frac{F(M + a, 1, M(1-p_0) + c, 1 - p_n(\path))}
{F(M + 1, 1, M(1-p_0) + 1, 1 - p_n(\path))} = 1.
\end{align*}
\end{lemme}
\begin{lemme}
\label{p_conditional}
Let $\path' \in \Pi$. 
For all $\path \in \Pi$ such that $p_n(\path) > p_{0}$, we have
\[
\lim  \limits_{M \to \infty} \P \big( \hat{p}_{M,n}( \path') > p_{0} 
\big | \hat{p}_{M,n}(\path) > p_{0}, \mathscr{D}_{n} \big) = \mathds{1}_{p_n ( \path' ) > p_{0}}
\]
and
\[
\lim  \limits_{M \to \infty} \P \big( \hat{p}_{M,n}( \path') > p_{0} 
\big| \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n} \big)= \mathds{1}_{\substack{p_n(\path') > p_{0} -  \rho_{n}(\path, \path')
\frac{\sigma_{n}( \path')}{\sigma_{n}(\path) } \\ \quad \times (p_{0}
 - p_n(\path))}}.
\]
Symmetrically, for all $\path \in \Pi$ such that $p_n(\path) \leq p_{0}$, we have
\begin{align*}
\lim  \limits_{M \to \infty} \P \big( \hat{p}_{M,n}( \path') > p_{0}
\big | \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n}  \big)
& = \mathds{1}_{p_n ( \path' ) > p_{0} },\\
\lim  \limits_{M \to \infty} \P \big( \hat{p}_{M,n}( \path') > p_{0}
\big | \hat{p}_{M,n}(\path) > p_{0}, \mathscr{D}_{n}  \big)
& = \mathds{1}_{\substack{p_n(\path') > p_{0} - \rho_{n}(\path, \path')
\frac{\sigma_{n}( \path' )}{\sigma_{n}(\path) } \\ \quad \times (p_{0} -p_n(\path)) }}.
 \end{align*}
\end{lemme}
We are now in a position to prove Theorem \ref{stability_equivalent}.
\begin{proof}[Proof of Theorem \ref{stability_equivalent}]
The first statement, identity (\ref{eq_stab_M}), is proved similarly to Corollary 2, using the law of large numbers instead of Theorem \ref{theorem_consistency}. For the second statement, we first recall that, by definition, 
\begin{align*}
\hat{S}_{M_{n},n,p_{0}} & =\frac{2\underset{\path \in \Pi}{\sum}\mathds{1}_{\hat{p}_{M_{n},n}(\path)>p_{0}
\cap\hat{p}_{M_{n},n}'(\path)>p_{0}}}{\underset{\path\in\Pi}{\sum}
\mathds{1}_{\hat{p}_{M_{n},n}(\path)>p_{0}}+\mathds{1}_{\hat{p}_{M_{n},n}'(\path)>p_{0}}} \\
& =1-\frac{\protect\underset{\path\in\Pi}{\sum}\mathds{1}_{\hat{p}_{M,n}(\path)>p_{0}\cap\hat{p}_{M,n}'(\path)\leq p_{0}}+\mathds{1}_{\hat{p}_{M,n}(\path)\leq p_{0}\cap\hat{p}_{M,n}'(\path)>p_{0}}}{\protect\underset{\path\in\Pi}{\sum}\mathds{1}_{\hat{p}_{M,n}(\path)>p_{0}}+\mathds{1}_{\hat{p}_{M,n}'(\path)>p_{0}}}.
\end{align*}
Taking the expectation conditional on $\mathscr{D}_{n}$ gives
\begin{align*}
\mathbb{E}\big[\hat{S}_{M,n,p_{0}} \big| \mathscr{D}_{n}  \big] & =1-2 ~\mathbb{E}\Bigg[\frac{\underset{\path\in\Pi}{\sum}\mathds{1}_{\hat{p}_{M,n}(\path)>p_{0}\cap\hat{p}_{M,n}'(\path)\leq p_{0}}}{\underset{\path\in\Pi}{\sum}\mathds{1}_{\hat{p}_{M,n}(\path)>p_{0}}+\mathds{1}_{\hat{p}_{M,n}'(\path)>p_{0}}} \Bigg| \mathscr{D}_{n}  \Bigg]  \\
& =1-2~\mathbb{E}\Big[\frac{U_{M}}{V_{M}+V_{M}'} \big| \mathscr{D}_{n}  \Big],
\end{align*}
where
$U_{M}=\underset{\path\in\Pi}{\sum}\mathds{1}_{\hat{p}_{M,n}(\path)>p_{0}\cap\hat{p}_{M,n}'(\path)\leq p_{0}}$,
$V_{M}=\underset{\path\in\Pi}{\sum}\mathds{1}_{\hat{p}_{M,n}(\path)>p_{0}}$, and $V_{M}'=\underset{\path\in\Pi}{\sum}\mathds{1}_{\hat{p}_{M,n}'(\path)>p_{0}}$.
Note that 
\begin{align*}
\mathbb{E}[V_{M} | \mathscr{D}_{n} ] & =\underset{\path\in\Pi}{\sum}\mathbb{P}(\hat{p}_{M,n}(\path)>p_{0}
 | \mathscr{D}_{n})\underset{M\rightarrow\infty}{\longrightarrow}\underset{\path\in\Pi}{\sum}\mathds{1}_{p_n(\path)>p_{0}}, \\
 \mathbb{E}[U_{M} | \mathscr{D}_{n}] & =\underset{\path\in\Pi}{\sum}\mathbb{P}(\hat{p}_{M,n}(\path)>p_{0}
 | \mathscr{D}_{n} )\mathbb{P}(\hat{p}_{M,n}(\path)\leq p_{0} | \mathscr{D}_{n} )\underset{M\rightarrow\infty}{\longrightarrow}0.
\end{align*}
Also, 
\begin{align*}
\mathbb{E}\Big[ \frac{U_{M}}{V_{M}+V_{M}'} & \big| \mathscr{D}_{n}  \Big] \\
 = \sum_{m,m'} & \frac{1}{m+m'}\mathbb{E}[U_{M}|V_{M}=m,V_{M}'=m' , \mathscr{D}_{n}] \\ & \times \mathbb{P}(V_{M}=m | \mathscr{D}_{n} )\mathbb{P}(V_{M}'=m' | \mathscr{D}_{n} )\\
 =\sum_{m,m'} & \frac{1}{m+m'}\mathbb{E}\Big[\underset{\path\in\Pi}{\sum}\mathds{1}_{\hat{p}_{M,n}(\path)>p_{0}\cap\hat{p}_{M,n}'(\path)\leq p_{0}}\big|V_{M}=m,V_{M}'=m' , \mathscr{D}_{n} \Big] \\ & \times \mathbb{P}(V_{M}=m | \mathscr{D}_{n})\mathbb{P}(V_{M}'=m' | \mathscr{D}_{n} ) \\
 =\sum_{m,m'}&\frac{1}{m+m'}\underset{\path\in\Pi}{\sum}\mathbb{P}(\hat{p}_{M,n}(\path)>p_{0}|V_{M}=m , \mathscr{D}_{n} ) \\ \times \mathbb{P}&(\hat{p}_{M,n}'(\path)\leq p_{0}|V_{M}'=m' , \mathscr{D}_{n} ) \mathbb{P}(V_{M}=m | \mathscr{D}_{n})\mathbb{P}(V_{M}'=m' | \mathscr{D}_{n} )\\
=\sum_{m,m'}&\frac{1}{m+m'}\underset{\path\in\Pi}{\sum}\mathbb{P}(\hat{p}_{M,n}(\path)>p_{0},V_{M}=m | \mathscr{D}_{n} ) \\ & \times \mathbb{P}(\hat{p}_{M,n}(\path)\leq p_{0},V_{M}'=m' | \mathscr{D}_{n} )\\
 =\underset{\path\in\Pi}{\sum}&\mathbb{P}(\hat{p}_{M,n}(\path)>p_{0} | \mathscr{D}_{n})
\mathbb{P}(\hat{p}_{M,n}(\path)\leq p_{0} | \mathscr{D}_{n} )\\
&\times \Big[\sum_{m,m'}\frac{1}{m+m'}\mathbb{P}(V_{M}=m|\hat{p}_{M,n}(\path)>p_{0} , \mathscr{D}_{n} ) \\ & \quad \times \mathbb{P}(V_{M}'=m'|\hat{p}_{M,n}(\path)\leq p_{0} , \mathscr{D}_{n})\Big].
\end{align*}
For all $\path\in\Pi$,
\begin{align*}
& \P (\hat{p}_{M,n}(\path)>p_{0} | \mathscr{D}_{n} )
 \P (\hat{p}_{M,n}(\path)\leq p_{0}| \mathscr{D}_{n} \big)\\
& \quad = \Phi( M p_{0}, M, p_n(\path)) ( 1 - \Phi( M p_{0}, M, p_n(\path) ) ),
\end{align*}
where $\Phi$ is the cdf of the binomial distribution. As a direct consequence of Lemma \ref{p_conditional}, 
\begin{align*}
&\lim \limits_{M \to \infty} \sum_{m, m'}  \frac{1}{m+m'}\mathbb{P}(V_{M}=m
|\hat{p}_{M,n}(\path)>p_{0}, \mathscr{D}_{n}) \\ 
& \qquad \times
\mathbb{P}(V_{M}=m'|\hat{p}_{M,n}(\path)\leq p_{0}, \mathscr{D}_{n} ) \\
& \quad = \frac{1}{\sum\limits_{\path' \in \Pi} \mathds{1}_{p_n ( \path' ) > p_{0}}
 + \mathds{1}_{p_n(\path') + \rho_{n}(\path, \path')
\frac{\sigma_{n}( \path' )}{\sigma_{n}(\path) } (p_{0} - p_n(\path)) > p_{0}}},
\end{align*}
which yields
\begin{align*}
& 1 - \mathbb{E} [ \hat{S}_{M,n,p_{0}} | \mathscr{D}_{n} ]\\ & \underset{M \to \infty}{\sim} \sum\limits_{\path \in \Pi}
\frac{2 \Phi( M p_{0}, M, p_n(\path)) ( 1 - \Phi( M p_{0}, M, p_n(\path) ) )}
{\sum\limits_{\path' \in \Pi} \mathds{1}_{\hat{p}_{n}(\path') > p_{0}} +
\mathds{1}_{p_n(\path') + \rho_{n}(\path, \path')
\frac{\sigma_{n}( \path' )}{\sigma_{n}(\path) } (p_{0} - p_n(\path)) > p_{0} }}.
\end{align*}
This is the desired result.
\end{proof}




\subsection{Proof of intermediate lemmas}

\begin{proof}[Proof of lemma \ref{F_limit}]
\cite{cvitkovic2017asymptotic} provides an asymptotic expansion of the hypergeometric function $F$ in the case where the first and third parameters
goes to infinity with a constant ratio. For $a, c, z, \varepsilon \in \R$, $b \notin \mathbb{Z} \setminus \mathbb{N}$, 
such that $\varepsilon > 1$, and $z \varepsilon < 1$,  \cite{cvitkovic2017asymptotic} gives in the section 2.2.2 (end of page 10)
\begin{align} \label{eq_F_expansion}
F(a + \varepsilon \lambda, b, c + \lambda, z)  \underset{|\lambda| \to \infty}{\sim} \frac{1}{(1 - \varepsilon z)^b}.
\end{align}
We can then derive the limit of the following ratio
\begin{align} \label{eq_F_ratio}
\lim \limits_{|\lambda| \to \infty} \frac{F(a + \varepsilon \lambda, b, c + \lambda, z)}{F(1 + \varepsilon \lambda, b, 1 + \lambda, z)} = 1
\end{align}
We use \ref{eq_F_ratio} in the specific case where $b = 1$, $a, c \in \mathbb{Z}$, $\varepsilon = \frac{1}{1 - p_0} > 1$, 
 $z = 1 - p_n(\path)$  for $\path \in \Pi$ such that $p_n(\path) > p_0$ (and then $z \varepsilon < 1$),
and $\lambda = M (1 - p_0)$, if follows that
\begin{align} \label{eq_FM_ratio}
\lim \limits_{M \to \infty} \frac{F(M + a , 1, M (1 - p_0) + c ,  1 - p_n(\path) )}
{F(M + 1, 1, M (1 - p_0) + 1,  1 - p_n(\path) )} = 1
\end{align}
\end{proof}

\begin{proof}[Proof of lemma \ref{p_conditional}]
Fix $\mathscr{D}_{n}$.  Let $\path', \path \in \Pi$. 
In what follows, when there is no ambiguity, we will replace $T(\Theta, \mathscr{D}_{n})$ by $T_n(\Theta)$ to lighten notations.


\paragraph{Case 1: $p_n(\path) > p_{0}$} 
\begin{align}
& \E\big[ \hat{p}_{M,n}\big( \path^{'} \big) | \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n}  \big] \nonumber \\
 = & \E\big[ \frac{1}{M}\sum_{l=1}^{M}\mathds{1}_{\path^{'}\in T_n(\Theta_{l})}
 ~\big|~ \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n}  \big] \nonumber \\
 = & \P \big( \path^{'} \in T_n(\Theta_{1})
 | \path \in T_n(\Theta_{1}), \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n}   \big) \nonumber \\
&  \times \P \big( \path \in T_n(\Theta_{1})  | \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n}  \big) \nonumber \\
 & +  \P \big( \path^{'} \in T_n(\Theta_{1})
 | \path \notin T_n(\Theta_{1}), \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n}  \big) \nonumber \\
 & \quad \times \big( 1 - \P \big( \path \in T_n(\Theta_{1})
 | \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n}  \big) \big) \nonumber \\
  = & \P \big( \path^{'} \in T_n(\Theta_{1})
 | \path \in T_n(\Theta_{1}), \mathscr{D}_{n}  \big)
\P \big( \path \in T_n(\Theta_{1})
 | \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n}  \big) \nonumber \\
 & +   \P \big( \path^{'} \in T_n(\Theta_{1})
 | \path \notin T_n(\Theta_{1}), \mathscr{D}_{n}  \big) \nonumber \\
& \quad \times \big( 1 - \P \big( \path \in T_n(\Theta_{1})
 | \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n}  \big) \big). \label{proof_thm2_eq1}
\end{align}
since
\begin{align}
 \P \big( \path^{'} \in T_n(\Theta_{1}) | \path \in T_n(\Theta_{1}), \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n}   \big) \nonumber \\
= \P \big( \path^{'} \in T_n(\Theta_{1}) | \path \in T_n(\Theta_{1}), \mathscr{D}_{n}  \big). \label{proof_thm2_eq2}
\end{align}
because, conditional on $\mathscr{D}_{n}$, the events  $\path^{'}\in T_n(\Theta_{1}), \hdots, \path^{'}\in T_n(\Theta_{M})$ are independent. 
We can rewrite, 
\begin{align}
& \P \big( \path^{'} \in T_n(\Theta_{1}) | \path \notin T_n(\Theta_{1}), \mathscr{D}_{n}  \big) \nonumber \\
& = \frac{\P \big( \path^{'} \in T_n(\Theta_{1}), \path \notin T_n(\Theta_{1})
| \mathscr{D}_{n}  \big)} {1 - p_n(\path)} \nonumber \\
& = \frac{\big( 1 - \P \big( \path \in T_n(\Theta_{1}) | \path^{'} \in T_n(\Theta_{1}), \mathscr{D}_{n}  \big) \big)
p_n\big( \path^{'} \big)}{1 - p_n(\path)} \nonumber \\
& = \frac{p_n\big( \path^{'} \big)}{1 - p_n(\path)}
- \frac{p_n(\path)}{1 - p_n(\path)}
\P \big( \path^{'}  \in T_n(\Theta_{1}) | \path \in T_n(\Theta_{1}), \mathscr{D}_{n}  \big), \label{proof_thm2_eq4}
\end{align}
yielding, using equation (\ref{proof_thm2_eq1}), 
\begin{align} \label{eq_mean_temp_1_0}
 \E \big[& \hat{p}_{M,n}\big( \path^{'} \big) | \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n} \big] \\
 = &\P \big( \path^{'} \in T_n(\Theta_{1})
 | \path \in T_n(\Theta_{1}), \mathscr{D}_{n}  \big)  \Big( \frac{\P \big( \path \in T_n(\Theta_{1})
 | \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n}  \big)}{1 -  p_n(\path)}  \nonumber \\
&-  \frac{p_n(\path)}{1 - p_n(\path)}\Big) + \frac{p_n\big( \path^{'} \big)}{1 - p_n(\path)}
\big( 1 - \P \big( \path \in T_n(\Theta_{1})  | \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n}  \big)  \big). \nonumber
\end{align}
Besides, by definition of the correlation
\begin{align*}
\rho_{n}\big(\path, \path^{'}\big) & = \frac{\textrm{Cov}\big(\mathds{1}_{\path \in T_n(\Theta)}, 
\mathds{1}_{\path^{'} \in T_n(\Theta)} | \mathscr{D}_{n}  \big)}
{\sigma_{n}(\path)\sigma_{n}\big( \path^{'} \big)},
 %& = \frac{ \P \big( \path^{'} \in T_n(\Theta_{1}), \path \in T_n(\Theta_{1}) \big| \mathscr{D}_{n}  \big) - p_n(\path) p_n\big( \path^{'} \big)} {\sigma_{n}(\path) \sigma_{n}\big( \path^{'} \big)}.
\end{align*}
simple calculations show that 
\begin{align}
\label{eq_mean_temp_2}
 \P & \big( \path^{'} \in T_n(\Theta_{1}) | \path \in T_n(\Theta_{1}),\mathscr{D}_{n}  \big) \nonumber \\
& =p_n\big( \path^{'} \big) +  \rho_{n}\big(\path, \path^{'}\big)
\sqrt{\frac{p_n\big( \path^{'} \big)}{p_n(\path)}
\big(1 - p_n(\path)\big)\big(1 -p_n\big( \path^{'} \big) \big)},
\end{align}
which, together with equation (\ref{eq_mean_temp_1_0}) leads to, 
\begin{align} \label{eq_mean_temp_1}
 \E &\big[ \hat{p}_{M,n}\big( \path^{'} \big) | \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n} \big] \\
&= p_n(\path^{'}) + \rho_{n}\big(\path, \path^{'}\big)
\frac{\sigma_{n}\big( \path^{'} \big)}{\sigma_{n}(\path) }
 \big(\P \big( \path \in T_n(\Theta_{1})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}  \big) \nonumber \\ & \hspace*{5.5cm} - p_n(\path)\big). \nonumber  
\end{align}
Regarding the probability in the right-hand side of equation (\ref{eq_mean_temp_1}), we have 
\begin{align*}
\P& \big( \path \in T_n(\Theta_{1})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}  \big) \\
& = p_n\big(\path \big) \frac{\P \big( \hat{p}_{M,n}(\path) \leq p_{0}
  | \path \in T_n(\Theta_{1}),\mathscr{D}_{n}  \big)}
{\P \big( \hat{p}_{M,n}(\path) \leq p_{0} | \mathscr{D}_{n}  \big)} \\
& =  p_n\big(\path \big) \frac{\P \big( (M-1) \hat{p}_{M-1,n}(\path) \leq M p_{0} - 1 | \mathscr{D}_{n}   \big)}
{\P \big( M \hat{p}_{M,n}(\path) \leq M p_{0} | \mathscr{D}_{n}   \big)} \\
& = p_n(\path) \frac{\Phi \big(Mp_0 - 1, M - 1, p_n(\path)\big)}
{\Phi \big(Mp_0, M, p_n(\path)\big)}.
\end{align*}
Using standard formulas, $\Phi$ can be expressed with the incomplete beta function, 
\begin{align*}
\Phi (k, M, p) = I_{1 - p}(M - k, k + 1) = \frac{B_{1-p}(M - k, k + 1)}{B(M - k, k + 1)},
\end{align*}
and the regularized beta function is related to the hypergeometric function $F$, for $a > 0$, $b > 0$, and $p \in [0,1]$
\citep{olver2010nist},
\begin{align*}
B_{1-p}(a,b) = \frac{(1-p)^ap^b}{a} F(a + b, 1, a + 1, 1 - p).
\end{align*}
Then, we can express the cdf of the binomial distribution using the hypergeometric function, and it follows
\begin{align} \label{eq_binom_ratio_0}
\P \big( \path& \in T_n(\Theta_{1})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}  \big) \\
&= p_{0} \frac{F(M, 1, M (1-p_{0}) + 1, 1 - \hat{p}_{n}\big(\path \big))}{F(M + 1, 1, M (1-p_{0}) + 1, 1 - \hat{p}_{n}\big(\path \big))} \nonumber
\end{align}
According to Lemma \ref{F_limit},
\begin{align} \label{eq_binom_ratio_5}
\lim \limits_{M \to \infty} \frac{F(M, 1, M (1-p_{0}) + 1, 1 - p_n(\path))}
{F(M + 1, 1, M (1-p_{0}) + 1, 1 - p_n(\path))} = 1.
\end{align}
Consequently,
\begin{align} \label{eq_binom_ratio_1}
\lim \limits_{M \to \infty} \P \big( \path \in T(\Theta_{1},\mathscr{D}_{n})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}  \big) = p_0,
\end{align}
and using this limiting result with equation (\ref{eq_mean_temp_1}) yields,  
\begin{align}
\label{eq_conditional_mean}
\lim \limits_{M \to \infty}  \E\big[ \hat{p}_{M,n}\big( \path^{'} \big)
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}  \big]
= p_n(\path^{'}) + &\rho_{n}\big(\path, \path^{'}\big)
\frac{\sigma_{n}\big( \path^{'} \big)}{\sigma_{n}(\path) } \\
 & \times \big(p_0 - p_n(\path)\big). \nonumber
\end{align}

Regarding the conditional variance, 
\begin{align*}
& \mathbb{V} \big[ \hat{p}_{M,n}\big( \path^{'} \big) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}  \big]\\
& = \mathbb{V} \big[ \frac{1}{M}\sum_{l=1}^{M}\mathds{1}_{\path^{'}\in T_n(\Theta_{l})}
 \big| \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}  \big] \\
& = \frac{1}{M} \mathbb{V} \big[ \mathds{1}_{\path^{'} \in T(\Theta_{1},\mathscr{D}_{n})}
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}  \big]\\
& \quad + (1 - \frac{1}{M})\textrm{Cov}(\mathds{1}_{\path^{'} \in T_n(\Theta_{1})},
\mathds{1}_{\path^{'} \in T_n(\Theta_{2})} | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n})\\
& \leq \frac{1}{M} + C_M
 \end{align*}
where
\begin{align*}
C_M & = \textrm{Cov}(\mathds{1}_{\path^{'} \in T_n(\Theta_{1})},
\mathds{1}_{\path^{'} \in T_n(\Theta_{2})} | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\
& = \P(\path^{'} \in T_n(\Theta_{1}), \path^{'} \in T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\
& \quad - \P(\path^{'} \in T_n(\Theta_{1}) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\
& \qquad  \times \P(\path^{'} \in T_n(\Theta_{2}) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\
\end{align*}


Then, we follow the same reasoning that leads to equation (\ref{eq_binom_ratio_1}). We can fully expand $C_M$ using Bayes formula, depending whether
$\path \in T_n(\Theta_{1})$ or $\path \in T_n(\Theta_{2})$. 
Note that, since all the trees are independent conditional on $\mathscr{D}_{n}$, we can reduce the conditioning event $\big\{\path \in T_n(\Theta_{1}),  \path \in T_n(\Theta_{2}), \hat{p}_{M,n}(\path) \leq p_{0}, \mathscr{D}_{n}\big\}$
 to $\big\{ \path \in T_n(\Theta_{1}),  \path \in T_n(\Theta_{2}), \mathscr{D}_{n}\big\}$, then
\begin{align*}
C_M  = \P( &\path^{'} \in T_n(\Theta_{1}), \path^{'} \in T_n(\Theta_{2})
 | \path \in T_n(\Theta_{1}), \path \in T_n(\Theta_{2}),\mathscr{D}_{n}) \\
& \times \P(\path \in T_n(\Theta_{1}), \path \in T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\
& - (\P(\path^{'} \in T_n(\Theta_{1}) | \path \in T_n(\Theta_{1}) ,\mathscr{D}_{n}) \\
& \qquad \times \P(\path \in T_n(\Theta_{1}) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}))^2 \\[5pt]
 + &2[ \P( \path^{'} \in T_n(\Theta_{1}), \path^{'} \in T_n(\Theta_{2})
 | \path \in T_n(\Theta_{1}), \path \notin T_n(\Theta_{2}),\mathscr{D}_{n}) \\
& \times \P(\path \in T_n(\Theta_{1}), \path \notin T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\
& - \P(\path^{'} \in T_n(\Theta_{1}) | \path \in T_n(\Theta_{1}) ,\mathscr{D}_{n}) \\
& \qquad \times \P(\path \in T_n(\Theta_{1}) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\
& \qquad \times \P(\path^{'} \in T_n(\Theta_{1}) | \path \notin T_n(\Theta_{1}) ,\mathscr{D}_{n}) \\
& \qquad \times \P(\path \notin T_n(\Theta_{1}) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) ] \\[5pt]
+ &\P( \path^{'} \in T_n(\Theta_{1}), \path^{'} \in T_n(\Theta_{2})
 | \path \notin T_n(\Theta_{1}), \path \notin T_n(\Theta_{2}),\mathscr{D}_{n}) \\
& \times \P(\path \notin T_n(\Theta_{1}), \path \notin T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\
& - (\P(\path^{'} \in T_n(\Theta_{1}) | \path \notin T_n(\Theta_{1}) ,\mathscr{D}_{n}) \\
& \qquad \times \P(\path \notin T_n(\Theta_{1}) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}))^2 \\
\end{align*}
Conditional on $\mathscr{D}_{n}$,  $T_n(\Theta_{1})$ and $ T_n(\Theta_{2})$ are independent, then
\begin{align*}
\P(& \path^{'} \in T_n(\Theta_{1}), \path^{'} \in T_n(\Theta_{2})
 | \path \in T_n(\Theta_{1}), \path \in T_n(\Theta_{2}),\mathscr{D}_{n}) \\
&= \frac{\P( \path^{'} \in T_n(\Theta_{1}), \path^{'} \in T_n(\Theta_{2}), \path \in T_n(\Theta_{1}), \path \in T_n(\Theta_{2})|\mathscr{D}_{n})}{\P(\path \in T_n(\Theta_{1}), \path \in T_n(\Theta_{2}) | \mathscr{D}_{n})}\\
&= \frac{\P( \path^{'} \in T_n(\Theta_{1}), \path \in T_n(\Theta_{1})|\mathscr{D}_{n}) \P(\path^{'} \in T_n(\Theta_{2}), \path \in T_n(\Theta_{2})|\mathscr{D}_{n})}{\P(\path \in T_n(\Theta_{1})|\mathscr{D}_{n}) \P( \path \in T_n(\Theta_{2})|\mathscr{D}_{n})} \\
&= \P( \path^{'} \in T_n(\Theta_{1}) | \path \in T_n(\Theta_{1}),\mathscr{D}_{n}) \P(\path^{'} \in T_n(\Theta_{2}) | \path \in T_n(\Theta_{2}),\mathscr{D}_{n})\\
&= \P( \path^{'} \in T_n(\Theta_{1}) | \path \in T_n(\Theta_{1}),\mathscr{D}_{n})^2
\end{align*}
we can rewrite $C_M$
\begin{align*}
 C_M  & = \P(\path^{'} \in T_n(\Theta_{1}) | \path \in T_n(\Theta_{1}) ,\mathscr{D}_{n})^2 \times \Delta_{M,1} \\
 & \quad + 2 \P(\path^{'} \in T_n(\Theta_{1}) | \path \in T_n(\Theta_{1}) ,\mathscr{D}_{n}) \\
& \qquad \times \P(\path^{'} \in T_n(\Theta_{1}) | \path \notin T_n(\Theta_{1}) ,\mathscr{D}_{n})  \times \Delta_{M,2} \\
 &  \quad + \P(\path^{'} \in T_n(\Theta_{1}) | \path \notin T_n(\Theta_{1}) ,\mathscr{D}_{n})^2  \times \Delta_{M,3},
\end{align*}
where
\begin{align*}
\Delta_{M,1} & = \P(\path \in T_n(\Theta_{1}), \path \in T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\
 & \quad - \P(\path \in T_n(\Theta_{1}) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n})^2, \\
 \Delta_{M,2} & = \P(\path \in T_n(\Theta_{1}), \path \notin T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\
& \quad - \P(\path \in T_n(\Theta_{1}) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\
& \qquad (1 -  \P(\path \in T_n(\Theta_{1}) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n})),\\
\Delta_{M,3} & = \P(\path \notin T_n(\Theta_{1}), \path \notin T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n})\\
 & \quad - \P(\path \notin T_n(\Theta_{1}) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n})^2.
\end{align*}
We first consider the term 
\begin{align*}
\Delta_{M,1} & = \P(\path \in T_n(\Theta_{1}), \path \in T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\ 
 & \quad - \P(\path \in T_n(\Theta_{1}) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n})^2
\end{align*}
Equation (\ref{eq_binom_ratio_1}) directly gives,
\begin{equation} \label{eq_binom_ratio_2}
\lim \limits_{M \to \infty} \P(\path \in T_n(\Theta_{1}) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n})^2
 = p_0^2.
\end{equation}
On the other hand
\begin{align*}
\P&(\path \in T_n(\Theta_{1}), \path \in T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\
&= p_n(\path)^2 \frac{\P( \hat{p}_{M,n}(\path) \leq p_{0} | \path \in T_n(\Theta_{1}), \path \in T_n(\Theta_{2}),\mathscr{D}_{n})}
{\P( \hat{p}_{M,n}(\path) \leq p_{0} | \mathscr{D}_{n})} \\
& = p_n(\path)^2 \frac{\Phi \big(Mp_0 - 2, M - 2, p_n(\path)\big)}
{\Phi \big(Mp_0, M, p_n(\path)\big)}.
\end{align*}
Again, as for equation (\ref{eq_binom_ratio_0}), we can express the cdf of the binomial distribution using the hypergeometric function $F$
\begin{align} \label{eq_binom_ratio_4} 
\P&(\path \in T_n(\Theta_{1}), \path \in T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \nonumber \\
& = p_0^2 \Big(1 + \frac{p_0 - 1}{p_0 (M-1)}\Big)  \frac{F(M-1, 1, M(1-p_0) + 1, 1 - p_n(\path))}{F(M+1, 1, M(1-p_0) + 1, 1 - p_n(\path))},
\end{align}
and from Lemma \ref{F_limit},
\begin{align*}
\lim \limits_{M \to \infty} \frac{F(M-1, 1, M(1-p_0) + 1, 1 - p_n(\path))}
{F(M+1, 1, M(1-p_0) + 1, 1 - p_n(\path))} = 1,
\end{align*}
that is 
\begin{align} \label{eq_binom_ratio_3}
\lim \limits_{M \to \infty} \P(\path \in T_n(\Theta_{1}), \path \in T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) = p_0^2.
\end{align}
Using equations (\ref{eq_binom_ratio_2}) and (\ref{eq_binom_ratio_3}), we conclude
\begin{align*}
\lim \limits_{M \to \infty} \Delta_{M,1} = 0.
\end{align*}
We follow the same reasoning for $\Delta_{M,3}$, equation (\ref{eq_binom_ratio_1}) gives
\begin{equation}
\lim \limits_{M \to \infty} \P(\path \notin T_n(\Theta_{1}) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n})^2
 = (1 - p_0)^2.
\end{equation}
On the other hand,
\begin{align*}
\P&(\path \notin T_n(\Theta_{1}), \path \notin T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\ 
 &= (1 - p_0)^2 \Big( 1 - \frac{p_0}{M-1} \Big) \frac{F(M-1, 1, M(1-p_0) -1 1, 1 - p_n(\path))}
{F(M+1, 1, M(1-p_0) + 1, 1 - p_n(\path))}
\end{align*}
From Lemma \ref{F_limit},
\begin{align}  \label{eq_binom_ratio_5}
\lim \limits_{M \to \infty} \P(\path \notin T_n(\Theta_{1}), \path \notin T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) = (1 - p_0)^2 
\end{align}
And finally $\lim \limits_{M \to \infty} \Delta_{M,3} = 0$. The term $\Delta_{M,2}$ can be treated in a similar way, since
equation (\ref{eq_binom_ratio_1}) gives
\begin{align*}
\lim \limits_{M \to \infty}& \P(\path \in T_n(\Theta_{1}) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n})
\P(\path \notin T_n(\Theta_{1}) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n})\\
&  = p_0(1 - p_0).    
\end{align*}
Simple identity shows 
\begin{align*}
& \P(\path \in T_n(\Theta_{1}), \path \notin T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\
 & = \frac{1}{2} \Big( 1 -  \P(\path \notin T_n(\Theta_{1}), \path \notin T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \\
& \quad - \P(\path \in T_n(\Theta_{1}), \path \in T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \Big).
\end{align*}
Taking the limit of the previous equation and using equations (\ref{eq_binom_ratio_3}) and (\ref{eq_binom_ratio_5}), we get
\begin{align}
\lim \limits_{M \to \infty}& \P(\path \in T_n(\Theta_{1}), \path \notin T_n(\Theta_{2})
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}) \nonumber\\  
 %& = \frac{1}{2} \big( 1 - p_0^2 - (1 - p_0)^2 \big) \\
 & = p_0 ( 1- p_0). \label{equation_proof_bis_bis}
\end{align}
Using (\ref{eq_binom_ratio_1}) and  (\ref{equation_proof_bis_bis}), $\lim \limits_{M \to \infty} \Delta_{M,2}= 0$. Since $\Delta_{M,1}, \Delta_{M,2}, \Delta_{M,3} \to 0$, we obtain $\lim \limits_{M \to \infty} C_M = 0$, that is, 
\begin{align}
\label{eq_conditional_variance}
\lim \limits_{M \to \infty}
\mathbb{V} \big[ \hat{p}_{M,n}\big( \path^{'} \big) | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n}  \big]
= 0.
\end{align}

Finally combining equations (\ref{eq_conditional_mean}) and (\ref{eq_conditional_variance}),
\begin{align*}
\lim  \limits_{M \to \infty} \P \big( \hat{p}_{M,n}\big( \path^{'} \big) > p_{0}
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n} \big) \\
= \mathds{1}_{p_n(\path^{'}) + \rho_{n}(\path, \path^{'})
\frac{\sigma_{n}( \path^{'} )}{\sigma_{n}(\path) }
 (p_{0} - p_n(\path)) > p_{0}} 
\end{align*} 



\paragraph{Case 2: $p_n(\path) \leq p_{0}$}
By the law of large numbers, $\lim \limits_{M \to \infty}  \hat{p}_{M,n}\big( \path \big) = p_n(\path) $ in probability, 
and consequently  $\lim \limits_{M \to \infty}  \P \big(\hat{p}_{M,n}\big( \path \big) \leq p_0 \big) = 1$.
Additionally, we can simply write
\begin{align*}
 \P \big( &\hat{p}_{M,n}\big( \path^{'} \big) > p_{0}
 | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n} \big) \\
&=  \frac{\P \big( \hat{p}_{M,n}\big( \path^{'} \big) > p_{0}, \hat{p}_{M,n}(\path) \leq p_{0} | \mathscr{D}_{n} \big)}
{\P \big(\hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n} \big)}
\end{align*} 
Again, by the law of large numbers, $\lim \limits_{M \to \infty}  \hat{p}_{M,n}\big(\path'\big) = p_n(\path') $ in probability.
Then, if $ p_n(\path') > p_0$, $\lim \limits_{M \to \infty}  \P \big(\hat{p}_{M,n}\big( \path' \big) > p_0 \big) = 1$,
and it follows that $\lim \limits_{M \to \infty} \P \big( \hat{p}_{M,n}( \path' ) > p_{0}, \hat{p}_{M,n}(\path) \leq p_{0} | \mathscr{D}_{n} \big) = 1$.
 If $ p_n(\path') \leq p_0$, $\lim \limits_{M \to \infty}  \P \big(\hat{p}_{M,n}\big( \path' \big) > p_0 \big) = 0$,
and consequently $\lim \limits_{M \to \infty} \P \big( \hat{p}_{M,n}\big( \path^{'} \big) > p_{0}, \hat{p}_{M,n}(\path) \leq p_{0} | \mathscr{D}_{n} \big) = 0$.
This can be compacted under the form
\begin{align*}
 \lim \limits_{M \to \infty} \P \big( \hat{p}_{M,n}\big( \path^{'} \big) > p_{0} | \hat{p}_{M,n}(\path) \leq p_{0},\mathscr{D}_{n} \big) = \mathds{1}_{p_n(\path') > p_0}.
\end{align*}

The proof for the case $\P \big[ \hat{p}_{M,n}\big( \path^{'} \big) > p_0 | \hat{p}_{M,n}(\path) > p_{0},\mathscr{D}_{n}  \big]$
is similar.
\end{proof}

\appendix

%\section*{Acknowledgements}

\bibliography{biblio}

\end{document}
