HARMONIC MOMENTS AND LARGE DEVIATIONS FOR A SUPERCRITICAL BRANCHING PROCESS IN A RANDOM ENVIRONMENT
Résumé
Let $(Z_n)$ be a supercritical branching process in an independent and identically distributed random environment $\xi$.
We study the asymptotic of the harmonic moments $\mathbb{E}\left[Z_n^{-r} | Z_0=k \right]$ of order $r>0$ as $n \to \infty$. We exhibit a phase transition with the critical value $r_k>0$
determined by the equation
$\mathbb E p_1^k = \mathbb E m_0^{-r_k},$ where $m_0=\sum_{k=0}^\infty k p_k$ with $p_k=\mathbb P(Z_1=k | \xi),$
assuming that $p_0=0.$
Contrary to the constant environment case (the Galton-Watson case), this critical value is different from that for the existence of the harmonic moments of $W=\lim_{n\to\infty} Z_n / \mathbb E (Z_n|\xi).$ The aforementioned phase transition is linked to that for the rate function of the lower large deviation for $Z_n$.
As an application, we obtain a lower large deviation result for $Z_n$ under weaker conditions than in previous works and give a new expression of the rate function.
We also improve an earlier result about the convergence rate in the central limit theorem for $W-W_n,$ and find an equivalence for the large deviation probabilities of the ratio $Z_{n+1} / Z_n$.
Domaines
Probabilités [math.PR]
Origine : Fichiers produits par l'(les) auteur(s)
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