Random walks with occasionally modified transition probabilities
Random walks with occasionally modified transition probabilities
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偶尔修改转移概率的随机游走
DOI：
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发表时间：
2009
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通讯作者：
Bruno Schapira
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作者：
Olivier Raimond;Bruno Schapira
本文研究了离散时间过程的常返性和（弱）大数定律的有效性，在最简单的情况下，离散时间过程是由$上的简单对称随机游动得到的。$通过从一个新的点修改一个步骤的分布。如果过程记为${S_n}_{n ge 0}$，则给定过去时间$n$的条件分布$S_{n+1} - S_n$是简单随机游动步的分布，只要$S_n$在$[0，n-1]$期间至少已经被访问过一次。因此，在这种情况下，$P{S_{n+1}-S_n = pm 1| S_ell，ell le n} = 1/2$。我们用$P_1$表示这个分布。然而，如果$S_n$在时间$n$之前没有被访问过的点，那么我们把$S_{n+1}-S_n$的条件分布，给定过去，取另一个分布$P_2$。我们想在特定的情况下决定$S_n$是否无限频繁地返回到原点，以及$（1/n）S_n是否 0$的概率。还考虑了$P_i$的概括或变体以及$P_i$之间的切换规则。
We study recurrence properties and the validity of the (weak) law of large numbers for (discrete time) processes which, in the simplest case, are obtained from simple symmetric random walk on $$ by modifying the distribution of a step from a fresh point. If the process is denoted as ${S_n}_{n ge 0}$, then the conditional distribution of $S_{n+1} - S_n$ given the past through time $n$ is the distribution of a simple random walk step, provided $S_n$ is at a point which has been visited already at least once during $[0,n-1]$. Thus in this case $P{S_{n+1}-S_n = pm 1|S_ell, ell le n} = 1/2$. We denote this distribution by $P_1$. However, if $S_n$ is at a point which has not been visited before time $n$, then we take for the conditional distribution of $S_{n+1}-S_n$, given the past, some other distribution $P_2$. We want to decide in specific cases whether $S_n$ returns infinitely often to the origin and whether $(1/n)S_n o 0$ in probability. Generalizations or variants of the $P_i$ and the rules for switching between the $P_i$ are also considered.