Theory of Probability &Its Applications, Volume 65, Issue 3, Page 359-374, January 2020.
Let $(p_i,q_i)$, $i\in {Z}$, 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 $i\in {Z}$. We consider a random walk in the random environment $\{(p_i,q_i)$, $i\in {Z}\}$. It is assumed that ${E}\ln (p_0/q_0)=0$ and $0<{E}\ln^{2}(q_0/p_0)<+\infty$. 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.

中文翻译：

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在随机环境中随机行走达到高水平的时代

概率论及其应用，第65卷，第3期，第359-374页，2020年1月。
令{Z} $中的$（p_i，q_i）$，$ i \是独立的相同分布的随机向量的序列，使得$ p_i，q_i> 0 $和$ p_i + q_i $ $ = 1 $ \ in {Z} $。我们考虑在随机环境$ \ {（p_i，q_i）$，{i}中的$ i \}中进行随机游走。假设$ {E} \ ln（p_0 / q_0）= 0 $和$ 0 <{E} \ ln ^ {2}（q_0 / p_0）<+ \ infty $。我们研究达到定额$ n $的水平$ n_1，\ dots，n_m $的水平达到$ T_ {n_1}，\ dots，T_ {n_m} $的时间。事实证明，可以将基础概率空间划分为随机事件（取决于$ n $），以使它们对大$ n $的概率接近于正数，并且在每个此类事件上，时间的集合都是$ T_ {n_1 }，\ dots，T_ {n_m} $分成连续的组，这样每个组的元素具有相同的顺序，并且与连续组的元素相比可以忽略不计。