Nonparametric regression estimation for random fields in a fixed-design

Mohamed EL MACHKOURI

4th February 2005

Abstract

We investigate the nonparametric estimation for regression in a fixed-design setting when the errors are given by a field of dependent random variables. Sufficient conditions for kernel estimators to converge uniformly are obtained. These estimators can attain the optimal rates of uniform convergence and the results apply to a large class of random fields which contains martingale-difference random fields and mixing random fields.

AMS Subject Classifications (2000): 60G60, 62G08 Key words and phrases: nonparametric regression estimation, kernel estimators, strong consistency, fixed-design, exponential inequalities, martingale difference random fields, mixing, Orlicz spaces. Short title: Nonparametric regression in a fixed design.

1 Introduction

Over the last few years nonparametric estimation for random fields (or spatial processes) was given increasing attention stimulated by a growing demand from applied research areas (see Guyon [18]). In fact, spatial data arise in various areas of research including econometrics, image analysis, meterology, geostatistics... Our aim in this paper is to investigate uniform strong convergence rates of a regression estimator in a fixed design setting when the errors are given by a stationary field of dependent random variables which show spatial interaction. We are most interested in conditions which ensure convergence rates to be identical to those in the case of independent errors (see Stone [33]). Currently the author is working on extensions of the present results to the random design framework. Let ℤ^{d}, d ≥ 1 denote the integer lattice points in the d-dimensional Euclidean space. By a stationary real random field we mean any family (ε_{k})_{k∈ℤ^{d}} of real-valued random variables defined on a probability space (Ω, F, ℙ) such that for any (k, n) ∈ ℤ^{d}×ℕ^{∗} and any (i_{1}, ..., i_{n}) ∈ (ℤ^{d})^{n}, the random vectors (ε_{i_{1}}, ..., ε_{i_{n}}) and (ε_{i_{1}+k}, ..., ε_{i_{n}+k}) have the same law. The regression model which we are interested in is

Y_{i} = g(i/n) + ε_{i}, i ∈ Λ_{n} = {1, ..., n}^{d}  (1)

where g is an unknown smooth function and (ε_{i})_{i∈ℤ^{d}} is a zero mean stationary real random field. Note that this model was considered also by Bosq [8] and Hall et Hart [19] for time series (d = 1). Let K be a probability kernel defined on ℝ^{d} and (h_{n})_{n≥1} a sequence of positive numbers which converges to zero and which satisfies (nh_{n})_{n≥1} goes to infinity. We estimate the function g by the kernel-type estimator g_{n} defined for any x in [0, 1]^{d} by

g_{n}(x) = \frac{∑_{i∈Λ_{n}} Y_{i} K (\frac{x−i/n}{h_{n}})}{∑_{i∈Λ_{n}} K (\frac{x−i/n}{h_{n}})}.  (2)

Note that Assumption A1) in section 2 ensures that g_{n} is well defined. Until now, most of existing theoretical nonparametric results of dependent random variables pertain to time series (see Bosq [9]) and relatively few generalizations to the spatial domain are available. Key references on this topic are Biau [5], Carbon et al. [10], Carbon et al. [11], Hallin et al. [20], [21], Tran [34], Tran and Yakowitz [35] and Yao [36] who have investigated nonparametric density estimation for random fields and Altman [2], Biau and Cadre [6], Hallin et al. [22] and Lu and Chen [25], [26] who have studied spatial prediction and spatial regression estimation. The classical asymptotic theory in statistics is built upon central limit theorems, law of large numbers and large deviations inequalities for the sequences of random variables. These classical limit theorems have been extended to the setting of spatial processes. In particular, some key results on the central limit theorem and its functional versions are Alexander and Pyke [1], Bass [3], Basu and Dorea [4], Bolthausen [7] and more recently Dedecker [12], [13], El Machkouri [16] and El Machkouri and Volný [17]. For a survey on limit theorems for spatial processes and some applications in statistical physics, one can refer to Nahapetian [28]. Note also that the main results (section 3) of this work are obtained via exponential inequalities for random fields discovered by El Machkouri [16]. The paper is organized as follows. The next section sets up the notations and the assumptions which will be considered in the sequel. In section 3, we present our main results on both weak and strong consistencies rates of the estimator g_{n}. The last section is devoted to the proofs.

2 Notations and Assumptions

In the sequel we denote ∥x∥ = max_{1≤k≤d} |x_{k}| for any x = (x_{1}, ..., x_{d}) ∈ [0, 1]^{d}. With a view to obtain optimal convergence rates for the estimator g_{n} defined by (2), we have to make the following assumptions on the regression function g and the probability kernel K:

A1) The probability kernel K is symmetric, nonnegative, supported by [−1, 1]^{d} and satisfies a Lipschitz condition |K(x) − K(y)| ≤ η∥x − y∥ for any x, y ∈ [−1, 1]^{d} and some η > 0. In addition there exists c, C > 0 such that c ≤ K(x) ≤ C for any x ∈ [−1, 1]^{d}.

A2) There exists a constant B > 0 such that |g(x) − g(y)| ≤ B∥x − y∥ for any x, y ∈ [0, 1]^{d}, that is g is B-Lipschitz.

A Young function ψ is a real convex nondecreasing function defined on ℝ^{+} which satisfies lim_{t→∞} ψ(t) = +∞ and ψ(0) = 0. We define the Orlicz space L_{ψ} as the space of real random variables Z defined on the probability space (Ω, F, ℙ) such that E[ψ(|Z|/c)] < +∞ for some c > 0. The Orlicz space L_{ψ} equipped with the so-called Luxemburg norm ∥.∥_{ψ} defined for any real random variable Z by

∥Z∥_{ψ} = inf{ c > 0 ; E[ψ(|Z|/c)] ≤ 1 }

is a Banach space. For more about Young functions and Orlicz spaces one can refer to Krasnosel’skii and Rutickii [24]. Let β > 0. We denote by ψ_{β} the Young function defined for any x ∈ ℝ^{+} by

ψ_{β}(x) = exp((x + ξ_{β})^{β}) − exp(ξ_{β}^{β}) where ξ_{β} = ((1 − β)/β)^{1/β} 11_{{0<β<1}}.

On the lattice ℤ^{d} we define the lexicographic order as follows: if i = (i_{1}, ..., i_{d}) and j = (j_{1}, ..., j_{d}) are distinct elements of ℤ^{d}, the notation i <_{lex} j means that either i_{1} < j_{1} or for some p in {2, 3, ..., d}, i_{p} < j_{p} and i_{q} = j_{q} for 1 ≤ q < p. Let the sets {V_{i}^{k}; i ∈ ℤ^{d} , k ∈ ℕ^{∗}} be defined as follows:

V_{i}^{1} = {j ∈ ℤ^{d}; j <_{lex} i},

and for k ≥ 2

V_{i}^{k} = V_{i}^{1} ∩ {j ∈ ℤ^{d}; |i − j| ≥ k} where |i − j| = max_{1≤l≤d} |i_{l} − j_{l}|.

For any subset Γ of ℤ^{d} define F_{Γ} = σ(ε_{i}; i ∈ Γ) and set

E_{|k|}(ε_{i}) = E(ε_{i}|F_{V_{i}^{|k|}}), k ∈ V_{i}^{1}.

Denote β(q) = 2q/(2−q) for 0 < q < 2 and consider the following conditions:

C1) ε_{0} ∈ L^{∞} and

∑_{k∈V_{0}^{1}} ∥ε_{k}E_{|k|}(ε_{0})∥_{∞} < ∞.

C2) There exists 0 < q < 2 such that ε_{0} ∈ L_{ψ_{β(q)}} and

∑_{k∈V_{0}^{1}} ∥\sqrt{|ε_{k}E_{|k|}(ε_{0})|}∥_{ψ_{β(q)}}^{2} < ∞.

C3) There exists p > 2 such that ε_{0} ∈ L^{p} and

∑_{k∈V_{0}^{1}} ∥ε_{k}E_{|k|}(ε_{0})∥_{\frac{p}{2}} < ∞.

C4) ε_{0} ∈ L^{2} and ∑_{k∈ℤ^{d}} |E(ε_{0}ε_{k})| < ∞.

Remark 1 Note that Dedecker [12] established the central limit theorem for any stationary square-integrable random field (ε_{k})_{k∈ℤ^{d}} which satisfies the condition ∑_{k∈V_{0}^{1}} ∥ε_{k}E_{|k|}(ε_{0})∥_{1} < ∞.

In classical statistical physics, there exists spatial processes which satisfy conditions C1),...,C4). For example, Nahapetian and Petrosian [29] gave sufficient conditions for a Gibbs field (ε_{k})_{k∈ℤ^{d}} to possess the following martingale difference property: for any i in ℤ^{d}, E(ε_{i}|F_{V_{i}^{1}}) = 0 a.s. Another examples of random fields which satisfy conditions C1),...,C4) can be found also among the class of mixing random fields. More precisely, given two sub-σ-algebras U and V of F, different measures of their dependence have been considered in the literature. We are interested by two of them. The α-mixing and ϕ-mixing coefficients had been introduced by Rosenblatt [31] and Ibragimov [23] respectively and can be defined by

α(U, V) = sup{|ℙ(U ∩ V) − ℙ(U)ℙ(V)|, U ∈ U, V ∈ V}
ϕ(U, V) = sup{∥ℙ(V |U) − ℙ(V)∥_{∞} , V ∈ V}.

We have 2α(U, V) ≤ ϕ(U, V) and these coefficients equal zero if and only if the σ-algebras U and V are independent. Denote by ♯Γ the cardinality of any subset Γ of ℤ^{d}. In the sequel, we shall use the following non-uniform mixing coefficients defined for any (k, l, n) in (ℕ^{∗} ∪ {∞})^{2} × ℕ by

α_{k,l}(n) = sup {α(F_{Γ_{1}}, F_{Γ_{2}}), ♯Γ_{1} ≤ k, ♯Γ_{2} ≤ l, ρ(Γ_{1}, Γ_{2}) ≥ n},
ϕ_{k,l}(n) = sup {ϕ(F_{Γ_{1}}, F_{Γ_{2}}), ♯Γ_{1} ≤ k, ♯Γ_{2} ≤ l, ρ(Γ_{1}, Γ_{2}) ≥ n},

where the distance ρ is defined by ρ(Γ_{1}, Γ_{2}) = min{|i − j|, i ∈ Γ_{1}, j ∈ Γ_{2}}. We say that the random field (ε_{k})_{k∈ℤ^{d}} is α-mixing or ϕ-mixing if there exists a pair (k, l) in (ℕ^{∗} ∪{∞})^{2} such that lim_{n→∞} α_{k,l}(n) = 0 or lim_{n→∞} ϕ_{k,l}(n) = 0 respectively. For more about mixing coefficients one can refer to Doukhan [15]. We consider the following mixing conditions:

C′1) ε_{0} ∈ L^{∞} and

∑_{k∈ℤ^{d}} ϕ_{∞,1}(|k|) < ∞.

C′2) There exists 0 < q < 2 such that ε_{0} ∈ L_{ψ_{β(q)}} and

∑_{k∈ℤ^{d}} \sqrt{ϕ_{∞,1}(|k|)} < ∞

or

∑_{k∈ℤ^{d}} c_{k}^{2}(β(q)) < ∞

where for any β > 0

c_{k}(β) = inf \{c > 0 | ∫_{0}^{α_{1,∞}(|k|)} ψ_{β} (\frac{Q_{ε_{0}}(u)}{c}) du ≤ 1\} .  (3)

C′3) There exists p > 2 such that ε_{0} ∈ L^{p} and

∑_{k∈ℤ^{d}} (∫_{0}^{α_{1,∞}(|k|)} Q_{ε_{0}}^{p}(u) du)^{2/p} < ∞  (4)

where Q_{ε_{0}} is the inverse cadlag of the tail function t → ℙ(|ε_{0}| > t) (i.e. for any u ≥ 0, Q_{ε_{0}}(u) = inf {t > 0 | ℙ(|ε_{0}| > t) ≤ u}).

Remark 2 Let us note that if p = 2 + δ for some δ > 0 then the condition

∑_{m=1}^{∞} m^{d−1}α_{1,∞}^{\frac{δ}{2+δ} −ε}(m) < ∞ for some ε > 0

is more restrictive than condition (4) and is known to be sufficient for the random field (ε_{k})_{k∈ℤ^{d}} to satisfy a functional central limit theorem (cf. Dedecker [13]).

In statistical physics, using the Dobrushin’s uniqueness condition (cf. [14]), one can construct Gibbs fields satisfying a uniform exponential mixing condition which is more restrictive than conditions C′1), C′2) and C′3) (see Guyon [18], theorem 2.1.3, p. 52).

3 Main results

Let (Z_{n})_{n≥1} be a sequence of real random variables and (v_{n})_{n≥1} be a sequence of positive numbers. We say that

Z_{n} = O_{a.s.} [v_{n}]

if there exists λ > 0 such that

lim sup_{n→∞} \frac{|Z_{n}|}{v_{n}} ≤ λ a.s.

Our main result is the following.

Theorem 1 Assume that the assumption A1) holds.

1) If C1) holds then

sup_{x∈[0,1]^{d}} |g_{n}(x) − Eg_{n}(x)| = O_{a.s.} [\frac{(log n)^{1/2}}{(nh_{n})^{d/2}} ] .  (5)

2) If C2) holds for some 0 < q < 2 then

sup_{x∈[0,1]^{d}} |g_{n}(x) − Eg_{n}(x)| = O_{a.s.} [\frac{(log n)^{1/q}}{(nh_{n})^{d/2}} ] .  (6)

3) Assume that C3) holds for some p > 2 and h_{n} = n^{−θ_{2}}(log n)^{θ_{1}} for some θ_{1}, θ_{2} > 0. Let a, b ≥ 0 be fixed and denote

v_{n} = \frac{n^{a}(log n)^{b}}{(nh_{n})^{d/2}} and θ = \frac{2a(d + p) − d^{2} − 2}{d(3d + 2)}.

If θ ≥ θ_{2} and d(3d + 2)θ_{1} + 2(d + p)b > 2 then
sup_{x∈[0,1]^{d}} |g_{n}(x) − Eg_{n}(x)| = O_{a.s.} [v_{n}] .  (7)

Remark 3 Theorem 1 shows that the optimal uniform convergence rate is obtained for bounded errors (cf. estimation (5)) and that it is “almost” optimal if one considers errors with only finite exponential moments (cf. estimation (6)).

Theorem 2 Assume that the assumption A1) holds.

1) Assume that C3) holds for some p > 2. Let a > 0 be fixed and denote

v_{n} = \frac{n^{a}}{(nh_{n})^{d/2}} and θ = \frac{2a(d + p) − d^{2}}{d(3d + 2)} .

If θ > 0 and h_{n} ≥ n^{−θ} then

∥ sup_{x∈[0,1]^{d}} |g_{n}(x) − Eg_{n}(x)|∥_{p} = O [v_{n}] .  (8)

2) If C4) holds then

sup_{x∈[0,1]^{d}} ∥g_{n}(x) − Eg_{n}(x)∥_{2} = O [(nh_{n})^{−d/2}] .  (9)

In the sequel, we denote by Lip(B) the set of B-Lipschitz functions. The following proposition gives the convergence of Eg_{n}(x) to g(x).

Proposition 1 Assume that the assumption A2) holds then

sup_{x∈[0,1]^{d}} sup_{g∈ Lip(B)} |Eg_{n}(x) − g(x)| = O [h_{n}] .

From Proposition 1 and Theorem 1 we derive the following corollary. Corollary 1 Assume that A1) and A2) hold and let h_{n} = (n^{−d} log n)^{1/(2+d)}.

1) If C1) holds then

sup_{x∈[0,1]^{d}} sup_{g∈ Lip(B)} |g_{n}(x) − g(x)| = O_{a.s.} [(\frac{log n}{n^{d}})^{\frac{1}{2+d}} ] .  (10)

2) If C2) holds for some 0 < q < 2 then

sup_{x∈[0,1]^{d}} sup_{g∈ Lip(B)} |g_{n}(x) − g(x)| = O_{a.s.} [u(n) (\frac{log n}{n^{d}})^{\frac{1}{2+d}} ]  (11)

where u(n) = (log n)^{(2−q)/2q}.

3) Let ε > 0 be fixed. If C3) holds for some p > 2 satisfying

p ≥ \frac{4d^{3} + (4 − 2ε)d^{2} + (2 − 4ε)d + 4}{2ε(2 + d)}  (12)

then

sup_{x∈[0,1]^{d}} sup_{g∈ Lip(B)} |g_{n}(x) − g(x)| = O_{a.s.} [u(n) (\frac{log n}{n^{d}})^{\frac{1}{2+d}} ]  (13)

where u(n) = n^{ε}. Remark 4 Note that the consistency rate (n^{−d} log n)^{1/(2+d)} is known to be the optimal one (see Stone [33]). From Proposition 1 and Theorem 2 we derive the following corollary. Corollary 2 Assume that A1) and A2) hold and let h_{n} = n^{−d/(2+d)}.

1) Let ε > 0 be fixed. If C3) holds for some p > 2 satisfying

p ≥ \frac{4d^{3} + (4 − 2ε)d^{2} − 4εd}{2ε(2 + d)}  (14)

then

∥ sup_{x∈[0,1]^{d}} sup_{g∈ Lip(B)} |g_{n}(x) − g(x)|∥_{p} = O [n^{− \frac{d}{2+d} +ε}] .  (15)

2) If C4) holds then

sup_{x∈[0,1]^{d}} ∥ sup_{g∈ Lip(B)} |g_{n}(x) − g(x)|∥_{2} = O [n^{− \frac{d}{2+d}} ] .  (16)

Finally the rates of convergence obtained above are valid when the errors are given by a mixing random field. More precisely, we have the following corollary.

Corollary 3 Theorems 1 and 2 and Corollaries 1 and 2 still hold if one replace conditions C1), C2) and C3) by conditions C′1), C′2) and C′3) respectively.

4 Proofs

For any x in [0, 1]^{d} and any integer n ≥ 1 we define B_{n}(x) = Eg_{n}(x) − g(x) and V_{n}(x) = g_{n}(x) − Eg_{n}(x). More precisely

B_{n}(x) = \frac{∑_{i∈Λ_{n}} a_{i}(x)g(i/n)}{∑_{i∈Λ_{n}} a_{i}(x)} − g(x)
V_{n}(x) = \frac{∑_{i∈Λ_{n}} a_{i}(x)ε_{i}}{∑_{i∈Λ_{n}} a_{i}(x)}

where a_{i}(x) = K (\frac{x−i/n}{h_{n}}). In the sequel, we denote also S_{n}(x) = ∑_{i∈Λ_{n}} a_{i}(x)ε_{i} for any x ∈ [0, 1]^{d}. We start with the following lemma.

Lemma 1 There exists constants c, C > 0 such that for any x ∈ [0, 1]^{d} and any n ∈ ℕ^{∗},

c ∏_{k=1}^{d}[n(x_{k} + h_{n})] ≤ ∑_{i∈Λ_{n}} a_{i}(x) ≤ C ∏_{k=1}^{d}[n(x_{k} + h_{n})]  (17)

where [ .] denote the integer part function.

Proof of Lemma 1. Since the kernel K is supported by [−1, 1]^{d}, we have

∑_{i∈Λ_{n}} a_{i}(x) = ∑_{i_{1}=1}^{[n(x_{1}+h_{n})]} … ∑_{i_{d}=1}^{[n(x_{d}+h_{n})]} a_{i}(x).

By assumption, there exists constants c, C > 0 such that c ≤ K(y) ≤ C for any y ∈ [−1, 1]^{d}. The proof of Lemma 1 is complete.

4.1 Proof of Theorem 1

Let (υ_{n})_{n≥1} be a sequence of positive numbers going to zero. Following Carbon and al. [11] the compact set [0, 1]^{d} can be covered by r_{n} cubes I_{k} having sides of length l_{n} = υ_{n}h_{n}^{2d+1} and center at c_{k}. Clearly there exists c > 0 such that r_{n} ≤ c/l_{n}^{d}. Define

A_{1,n}(g) = max_{1≤k≤r_{n}} sup_{x∈I_{k}} |g_{n}(x) − g_{n}(c_{k})|
A_{2,n}(g) = max_{1≤k≤r_{n}} sup_{x∈I_{k}} |Eg_{n}(x) − Eg_{n}(c_{k})|
A_{3,n} = max_{1≤k≤r_{n}} |g_{n}(c_{k}) − Eg_{n}(c_{k})|

then

sup_{x∈[0,1]^{d}} |g_{n}(x) − Eg_{n}(x)| ≤ sup_{g∈ Lip(B)}[A_{1,n}(g) + A_{2,n}(g)] + A_{3,n}. (18)

Lemma 2 For i = 1, 2 we have

sup_{g∈ Lip(B)} A_{i,n}(g) = O_{a.s.} [v_{n}] .

Proof of Lemma 2. Since g ∈ Lip(B), we can assume without loss of generality that g is bounded by B on the set [0, 1]^{d}. For any x ∈ I_{k}, we have

g_{n}(x) − g_{n}(c_{k}) = σ_{1} + σ_{2}

where

σ_{1} = \frac{∑_{i∈Λ_{n}} Y_{i}(a_{i}(x) − a_{i}(c_{k}))}{∑_{i∈Λ_{n}} a_{i}(x)}

and

σ_{2} = \frac{∑_{i∈Λ_{n}}(a_{i}(c_{k}) − a_{i}(x))}{∑_{i∈Λ_{n}} a_{i}(x) × ∑_{i∈Λ_{n}} a_{i}(c_{k})} ∑_{i∈Λ_{n}} Y_{i}a_{i}(c_{k}).

Now, by Lemma 1 and Assumption A1), we derive that there exists constants c, η > 0 such that for any n sufficiently large

|σ_{1}| ≤ \frac{2^{d}ηl_{n}/h_{n}}{c(nh_{n})^{d}} ∑_{i∈Λ_{n}} |Y_{i}| ≤ \frac{ηv_{n}h_{n}^{d}}{c} (B + \frac{1}{n^{d}} ∑_{i∈Λ_{n}} |ε_{i}|)

and

|σ_{2}| ≤ \frac{4^{d}ηn^{d}l_{n}/h_{n}}{c^{2}(nh_{n})^{2d}} ∑_{i∈Λ_{n}} |Y_{i}| ≤ \frac{ηv_{n}}{c^{2}} (B + \frac{1}{n^{d}} ∑_{i∈Λ_{n}} |ε_{i}|)

Since (ε_{i}) is a stationary ergodic random field the lemma easily follows from the last inequalities and the Birkhoff ergodic theorem. The proof of Lemma 2 is complete.

Lemma 3 Assume that either C1) holds and v_{n} = (log n)^{1/2}/(nh_{n})^{d/2} or C2) holds for some 0 < q < 2 and v_{n} = (log n)^{1/q}/(nh_{n})^{d/2} then

A_{3,n} = O_{a.s.}[v_{n}]

Proof of Lemma 3. Let 0 < q ≤ 2 be fixed. We consider the exponential Young function define for any x ∈ ℝ^{+} by ψ_{q}(x) = exp((x + ξ_{q})^{q}) − exp(ξ_{q}^{q}) where ξ_{q} = ((1 − q)/q)^{1/q} 11_{{0<q<1}}. Let λ > 0 and x ∈ [0, 1]^{d} be fixed

ℙ (|V_{n}(x)| > λv_{n}) = ℙ (|S_{n}(x)| > λv_{n} ∑_{i∈Λ_{n}} a_{i}(x))

≤ (1 + e^{ξ_{q}^{q}}) exp [ − (\frac{λv_{n} ∑_{i∈Λ_{n}} a_{i}(x)}{|| ∑_{i∈Λ_{n}} a_{i}(x)ε_{i}||_{ψ_{q}}} + ξ_{q})^{q} ].

For any i ∈ Λ_{n} and any 0 < q < 2 denote

b_{i,q}(a(x)ε) = ∥a_{i}(x)ε_{i}∥_{ψ_{β(q)}}^{2} + ∑_{k∈V_{i}^{1}} ∥\sqrt{|a_{k}(x)ε_{k}E_{|k−i|}(a_{i}(x)ε_{i})|}∥_{ψ_{β(q)}}^{2} (19)

and

b_{i,2}(a(x)ε) = ∥a_{i}(x)ε_{i}∥_{∞}^{2} + ∑_{k∈V_{i}^{1}} ∥a_{k}(x)ε_{k}E_{|k−i|}(a_{i}(x)ε_{i})∥_{∞}  (20)

where V_{i}^{1} = {j ∈ ℤ^{d}; j <_{lex} i}. Using Kahane-Khintchine inequalities (cf. El Machkouri [16], Theorem 1) we derive that if Condition C2) holds for some 0 < q < 2 then

ℙ (|V_{n}(x)| > λv_{n}) ≤ (1 + e^{ξ_{q}^{q}}) exp [ − (\frac{λ v_{n} ∑_{i∈Λ_{n}} a_{i}(x)}{M(∑_{i∈Λ_{n}} b_{i,q}(a(x)ε))^{1/2}} + ξ_{q})^{q} ]

(21)

where M is a positive constant depending only on q and on the probability kernel K. Now using the definition (19) and Lemma 1 there exist constants c, M > 0 such that

sup_{x∈[0,1]^{d}} ℙ (|V_{n}(x)| > λv_{n}) ≤ (1 + e^{ξ_{q}^{q}}) exp [ − (\frac{λ v_{n} (∑_{i∈Λ_{n}} a_{i}(x))^{1/2}}{M} + ξ_{q})^{q} ]

≤ (1 + e^{ξ_{q}^{q}}) exp [ − \frac{c^{q} λ^{q} v_{n}^{q} ([nh_{n}])^{dq/2}}{M^{q}} ]

So if v_{n} = (log n)^{1/q}/(nh_{n})^{d/2} and n is sufficiently large then

sup_{x∈[0,1]^{d}} ℙ (|V_{n}(x)| > λv_{n}) ≤ (1 + e^{ξ_{q}^{q}}) exp [ − \frac{c^{q} λ^{q} log n}{2^{dq/2}M^{q}}].  (22)

If Condition C1) holds then (21) still hold with q = 2 (cf. El Machkouri [16], Theorem 1). So if v_{n} = (log n)^{1/2}/(nh_{n})^{d/2} and n is large it follows that

sup_{x∈[0,1]^{d}} ℙ (|V_{n}(x)| > λv_{n}) ≤ 2 exp [ − \frac{c^{2} λ^{2} log n}{2^{d}M^{2}}].  (23)

Since

ℙ (|A_{3,n}| > λv_{n}) ≤ r_{n} sup_{x∈[0,1]^{d}} ℙ (|V_{n}(x)| > λv_{n}) ,

using (22) and (23), choosing λ sufficiently large and applying Borel-Cantelli’s lemma, we derive

ℙ (lim sup_{n→∞}{|A_{3,n}| > λv_{n}}) = 0

and

ℙ (lim sup_{n→∞} \frac{|A_{3,n}|}{v_{n}} ≤ λ) = 1.

The proof of points 1) and 2) of Theorem 1 are completed by combining Inequality (18) with Lemmas 2 and 3.

Lemma 4 Assume that C3) holds for some p > 2 and h_{n} = n^{−θ_{2}}(log n)^{θ_{1}} for some θ_{1}, θ_{2} > 0. Let a, b ≥ 0 be fixed and denote

v_{n} = \frac{n^{a}(log n)^{b}}{(nh_{n})^{d/2}} and θ = \frac{2a(d + p) − d^{2} − 2}{d(3d + 2)}.

If θ ≥ θ_{2} and d(3d + 2)θ_{1} + 2(d + p)b > 2 then

lim_{n→+∞} \frac{|A_{3,n}|}{v_{n}} = 0 a.s.

Proof of Lemma 4. Let p > 2 be fixed. For any λ > 0

ℙ (|V_{n}(x)| > λv_{n}) = ℙ (|S_{n}(x)| > λv_{n} ∑_{i∈Λ_{n}} a_{i}(x))

≤ \frac{λ^{−p}E|S_{n}(x)|^{p}}{v_{n}^{p}(∑_{i∈Λ_{n}} a_{i}(x))^{p}}

≤ \frac{λ^{−p}}{v_{n}^{p}(∑_{i∈Λ_{n}} a_{i}(x))^{p}} (2p ∑_{i∈Λ_{n}} c_{i}(x))^{p/2}

where c_{i}(x) = a_{i}(x)^{2}∥ε_{i}∥_{p}^{2} + a_{i}(x) ∑_{k∈V_{i}^{1}} a_{k}(x)∥ε_{k}E_{|k−i|}(ε_{i})∥_{\frac{p}{2}} . The last estimate follows from a Marcinkiewicz-Zygmund type inequality by Dedecker (see [13]) for real random fields. Noting that there exists γ > 0 such that c_{i}(x) ≤ γa_{i}(x), x ∈ [0, 1]^{d} and using Lemma 1, we derive that there exists γ′ > 0 such that

ℙ (|A_{3,n}| > λv_{n}) ≤ r_{n} sup_{x∈[0,1]^{d}} ℙ (|V_{n}(x)| > λv_{n}) ≤ \frac{γ′}{τ_{n}λ^{p}}

where τ_{n} = l_{n}^{d}v_{n}^{p}([nh_{n}])^{dp/2}. Since v_{n} = n^{a}(log n)^{b}/(nh_{n})^{d/2} and l_{n} = v_{n}h_{n}^{2d+1} it follows

\frac{1}{τ_{n}} = \frac{(nh_{n})^{d(d+p)/2}}{h_{n}^{d(2d+1)}n^{a(d+p)}(log n)^{b(d+p)}([nh_{n}])^{dp/2}} .

If n is sufficiently large, we derive

\frac{1}{τ_{n}} ≤ \frac{2^{dp/2}(nh_{n})^{d(d+p)/2}}{h_{n}^{d(2d+1)}n^{a(d+p)}(log n)^{b(d+p)}(nh_{n})^{dp/2}}

= \frac{2^{dp/2}}{h_{n}^{d(3d+2)/2}n^{a(d+p)−d^{2}/2}(log n)^{b(d+p)}}

≤ \frac{2^{dp/2}}{n(log n)^{b(d+p)+θ_{1}d(3d+2)/2}} since θ ≥ θ_{2}.

Now b(d + p) + θ_{1}d(3d + 2)/2 > 1 implies ∑_{n≥1} τ_{n}^{−1} < ∞. Applying Borel-Cantelli’s lemma, it follows that for any λ > 0

ℙ (lim sup_{n→∞}{|A_{3,n}| > λv_{n}}) = 0,

that is for any λ > 0

ℙ (lim sup_{n→∞} \frac{|A_{3,n}|}{v_{n}} ≤ λ) = 1.

The proof of Lemma 4 is complete and the point 3) of Theorem 1 is obtained by combining Inequality (18) with Lemmas 2 and 4. The proof of Theorem 1 is complete.

4.2 Proof of Theorem 2

We follow the first part of the proof of Theorem 1 and we consider the estimation (18).

Lemma 5 Assume that C3) holds for some p > 2. Let a > 0 be fixed and denote

v_{n} = \frac{n^{a}}{(nh_{n})^{d/2}} and θ = \frac{2a(d + p) − d^{2}}{d(3d + 2)} .

If θ > 0 and h_{n} ≥ n^{−θ} then

∥A_{3,n}∥_{p} = O [v_{n}] .

Proof of Lemma 5. Let p > 2 and x ∈ [0, 1]^{d} be fixed. Using the Marcinkiewicz-Zygmund type inequality by Dedecker (see [13]) as in the proof of Lemma 4 there exist γ′′, c > 0 such that

∥V_{n}(x)∥_{p} = (\frac{E|S_{n}(x)|^{p}}{(∑_{i∈Λ_{n}} a_{i}(x))^{p}} )^{1/p}

≤ γ′′ (∑_{i∈Λ_{n}} a_{i}(x))^{−1/2}

≤ \frac{γ′′}{\sqrt{c}}([nh_{n}])^{−d/2} by Lemma 1.

It follows that

r_{n}^{1/p} sup_{x∈[0,1]^{d}} ∥V_{n}(x)∥_{p} = O [\frac{v_{n}}{τ_{n}} ]

where τ_{n} = l_{n}^{d/p}v_{n}([nh_{n}])^{d/2}. If n is sufficiently large then τ_{n} ≥ 2^{−d/2}l_{n}^{d/p}v_{n}(nh_{n})^{d/2}, hence using h_{n} ≥ n^{−θ} we obtain τ_{n} ≥ 2^{−d/2}. Finally, we derive

∥A_{3,n}∥_{p} = ∥ max_{1≤k≤r_{n}} |V_{n}(x_{k})|∥_{p} ≤ r_{n}^{1/p} sup_{x∈[0,1]^{d}} ∥V_{n}(x)∥_{p} = O [v_{n}] .

The proof of Lemma 5 is complete. The point 1) of Theorem 2 is obtained by combining inequality (18) and lemmas 2 and 5.

Now, we are going to prove the point 2) of Theorem 2. We have

E(S_{n}(x)^{2}) = ∑_{k,l∈Λ_{n}} a_{k}(x) a_{l}(x)E(ε_{k}ε_{l})

= ∑_{k∈Λ_{n}} a_{k}(x)^{2}E(ε_{k}^{2}) + ∑_{k≠l} a_{k}(x)a_{l}(x)E(ε_{k}ε_{l})

= E(ε_{0}^{2}) ∑_{k∈Λ_{n}} a_{k}(x)^{2} + ∑_{k∈Λ_{n}} a_{k}(x) ∑_{l∈Λ_{n}\{k}} a_{l}(x)E(ε_{k}ε_{l})

≤ ∑_{l∈ℤ^{d}}|E(ε_{0}ε_{l})| × ∑_{k∈Λ_{n}} a_{k}(x).

If Condition C4) holds then using Lemma 1 there exists γ > 0 such that for any x ∈ [0, 1]^{d} we have E(S_{n}(x)^{2}) ≤ γ ∏_{k=1}^{d}[n(x_{k} + h_{n})]. Let x ∈ [0, 1]^{d} be fixed, using Lemma 1, there exists c > 0 such that

∥V_{n}(x)∥_{2} = \frac{∥S_{n}(x)∥_{2}}{∑_{i∈Λ_{n}} a_{i}(x)}

≤ \frac{\sqrt{γ}}{c} (∏_{k=1}^{d}[n(x_{k} + h_{n})])^{−1/2}

≤ \frac{\sqrt{γ}}{c ([nh_{n}])^{d/2}}

≤ \frac{2^{d/2}\sqrt{γ}}{c (nh_{n})^{d/2}} for n sufficiently large.

The proof of Theorem 2 is complete.

4.3 Proof of Proposition 1

Since g ∈ Lip(B), it follows that

|B_{n}(x)| = |\frac{∑_{i∈Λ_{n}}(g(i/n) − g(x))a_{i}(x)}{∑_{i∈Λ_{n}} a_{i}(x)}|

≤ Bh_{n} \frac{∑_{i∈Λ_{n}} ∥(i/n − x)/h_{n}∥a_{i}(x)}{∑_{i∈Λ_{n}} a_{i}(x)}

≤ Bh_{n}.

The proof of Proposition 1 is complete.

4.4 Proof of Corollary 1

Let h_{n} = (n^{−d} log n)^{1/(2+d)} then Proposition 1 gives

sup_{x∈[0,1]^{d}} sup_{g∈ Lip(B)} |Eg_{n}(x) − g(x)| = O [(\frac{log n}{n^{d}})^{\frac{1}{2+d}} ] .  (24)

Assume that C1) holds. Noting that

\frac{(log n)^{1/2}}{(nh_{n})^{d/2}} = (\frac{log n}{n^{d}})^{\frac{1}{2+d}}

and using (5) we obtain

sup_{x∈[0,1]^{d}} |g_{n}(x) − Eg_{n}(x)| = O_{a.s.} [(\frac{log n}{n^{d}})^{\frac{1}{2+d}} ] .  (25)

Combining (24) and (25) we derive (10).

Assume that C2) holds for some 0 < q < 2. Noting that

\frac{(log n)^{1/q}}{(nh_{n})^{d/2}} = (\frac{log n}{n^{d}})^{\frac{1}{2+d}} × (log n)^{(2−q)/2q}

and using (6) we obtain

sup_{x∈[0,1]^{d}} |g_{n}(x) − Eg_{n}(x)| = O_{a.s.} [(\frac{log n}{n^{d}})^{\frac{1}{2+d}} × (log n)^{(2−q)/2q}] .  (26)

Combining (24) and (26) we derive (11).

Let ε > 0 be fixed and assume that C3) holds for some p > 2 which satisfies condition (12). Applying the point 3) of Theorem 1 with θ_{1} = 1/(2 + d) and θ_{2} = d/(2 + d) and noting that

v_{n} = \frac{n^{a}(log n)^{b}}{(nh_{n})^{d/2}} = n^{ε} (\frac{log n}{n^{d}})^{\frac{1}{2+d}} ⟺ \{a = ε and b = \frac{1}{2}\}

it follows

sup_{x∈[0,1]^{d}} |g_{n}(x) − Eg_{n}(x)| = O_{a.s.} [n^{ε} (\frac{log n}{n^{d}})^{\frac{1}{2+d}} ] .  (27)

Combining (24) and (27) we derive (13). The proof of Corollary 1 is complete.

4.5 Proof of Corollary 2

Let h_{n} = n^{−d/(2+d)} then Proposition 1 gives

sup_{x∈[0,1]^{d}} sup_{g∈ Lip(B)} |Eg_{n}(x) − g(x)| = O [n^{− \frac{d}{2+d}} ] .  (28)

Let ε > 0 be fixed and assume that C3) holds for some p > 2 which satisfies condition (14). Applying the point 1) of Theorem 2 and noting that

v_{n} = \frac{n^{a}}{(nh_{n})^{d/2}} = n^{− \frac{d}{2+d} +ε} ⟺ a = ε

it follows that

∥ sup_{x∈[0,1]^{d}} |g_{n}(x) − Eg_{n}(x)|∥_{p} = O [n^{− \frac{d}{2+d} +ε}] .  (29)

Combining (28) and (29) we derive (15).

Since h_{n} = n^{−d/(2+d)} then (nh_{n})^{−d/2} = h_{n}. So, if C4) holds then combining (28) and (9) we derive (16). The proof of Corollary 2 is complete.

4.6 Proof of Corollary 3

Let p > 2 be fixed. Using Rio’s inequality [30] (see also Dedecker [13]) we obtain the bound

∥ε_{k}E_{|k|}(ε_{0})∥_{\frac{p}{2}} ≤ 4 (∫_{0}^{α_{1,∞}(|k|)} Q_{ε_{0}}^{p}(u) du)^{2/p}  (30)

hence condition C′3) is more restrictive than condition C3). By Serfling’s inequality (see McLeish [27] or Serfling [32]) we know that

∥ε_{k}E_{|k|}(ε_{0})∥_{∞} ≤ 2∥ε_{0}∥_{∞}^{2}ϕ_{∞,1}(|k|)

so condition C′1) is more restrictive than condition C1). Now for 0 < q < 2 there exists C(q) > 0 (cf. Inequality (17) in [16]) such that

∥\sqrt{|ε_{k}E_{|k|}(ε_{0})|}∥_{ψ_{β(q)}}^{2} ≤ C(q)\sqrt{ϕ_{∞,1}(|k|)}.  (31)

In [16] we used the following lemma which can be obtain by the expansion of the exponential function.

Lemma 6 Let β be a positive real number and Z be a real random variable. There exist positive universal constants A_{β} and B_{β} depending only on β such that

A_{β} sup_{p>2} \frac{∥Z∥_{p}}{p^{1/β}} ≤ ∥Z∥_{ψ_{β}} ≤ B_{β} sup_{p>2} \frac{∥Z∥_{p}}{p^{1/β}} .

Consider the coefficient c_{k}(β) given by (3) and denote

d_{k}(p) = (∫_{0}^{α_{1,∞}(|k|)} Q_{ε_{0}}^{p}(u) du)^{1/p}

then the following version of lemma 6 holds.

Lemma 7 Let β be a positive real number. There exist positive universal constants A_{β} and B_{β} depending only on β such that for any k ∈ ℤ^{d}

A_{β} sup_{p>2} \frac{d_{k}(p)}{p^{1/β}} ≤ c_{k}(β) ≤ B_{β} sup_{p>2} \frac{d_{k}(p)}{p^{1/β}}.

Now combining lemmas 6 and 7 and inequality (30) there exists C′(q) > 0 such that

∥\sqrt{|ε_{k}E_{|k|}(ε_{0})|}∥_{ψ_{β(q)}}^{2} ≤ C′(q) c_{k}^{2}(β(q)).  (32)

Finally condition C′2) is more restrictive than condition C2) and the proof of Corollary 3 is complete.

Aknowledgements. I would like to express my thanks to the anonymous referee for his/her careful reading of the manuscript and valuable suggestions. I am indebted for É. Youndje for many stimulating conversations on nonparametric estimation.

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Mohamed EL MACHKOURI Laboratoire de Mathématiques Raphaël Salem UMR 6085, Université de Rouen Site Colbert, F76821 Mont-Saint-Aignan Cedex email : mohamed.elmachkouri@univ-rouen.fr