On the times of attaining high levels by a random walk in a random environment

V. I. Afanasyev

Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 3, pp. 460-478

Résumé

Let $(p_i,q_i)$, $i\in \mathbf{Z}$, be a sequence of independent identically distributed random vectors such that $p_i,q_i>0$ and $p_i+q_i$ $=1$ a.s. for $i\in \mathbf{Z}$. We consider a random walk in the random environment $\{(p_i,q_i)$, $i\in \mathbf{Z}\}$. It is assumed that $\mathbf{E}\ln (p_0/q_0)=0$ and $0\mathbf{E}\ln^{2}(q_0/p_0)+\infty$. We study the times of attaining $T_{n_1},\dots,T_{n_m}$ of increasing levels $n_1,\dots,n_m$ of order $n$. It is proved that the underlying probability space can be partitioned into random events (depending on $n$) such that their probabilities for large $n$ are close to positive numbers, and on each such event, the set of times $T_{n_1},\dots,T_{n_m}$ is partitioned into consecutive groups such that elements of each group have the same order and are negligible compared with those of the successive group.

Détail

Keywords: random walk in random environment, branching in random environment with immigration, limit theorem.