Local limit theorems and occupation times for perturbed random walks.

Abstract

Let $X\sb1 ,X\sb2 ,X\sb3 ,\...$ be a sequence of i.i.d. random variables with distribution function $F$, mean zero, variance 1. Let $\xi\sb1 ,\xi\sb2 ,\xi\sb3 ,\...$ be a sequence of random variables. Let $S\sb{n}$ = $X\sb1+X\sb2+\...+X\sb{n}$. Then $Z\sb{n}=S\sb{n}+\xi\sb{n}$ is called a perturbed random walk under certain restrictions on $\xi\sb{n}$. It is shown that under certain moment conditions on $\{X\sb{n}\}$, $\{\xi\sb{n}\}$ and certain variation conditions on $\{\xi\sb{n}\}$, the following hold: (i) a Stone type local limit theorem:$$\sqrt{n}\IP\{b < Z\sb{n}\le b+c\}\ \sbsp{n\to\infty}{\longrightarrow}\ {c\over\sqrt{2\pi}}$$for all $b\in\IR$, $c \in\IR\sp+$ = (0,$\infty$); (ii) a difference limit theorem for occupation times: Let$$N\sb{n}(J) = {1\over \sqrt{n}} \sum\limits\sbsp{k=1}{n} 1\sb{J}(Z\sb{k})$$be the normalized occupation time of $\{Z\sb{k}\}$ up to step $n$. Then$${N\sb{n}(J)\over\vert J\vert} - {N\sb{n}(K)\over\vert K\vert} {\sbsp{\longrightarrow}{\cal P}\atop n\to\infty}\ 0$$where $J,K$ are any non-degenerate, finite intervals on $\IR$, where $\vert J\vert$, $\vert K\vert$ are the lengths of $J$ and $K$. (iii) a Kallianpur-Robbins theorem:$${1\over c}N\sb{n}((0,c\rbrack)\ \sbsp{\longrightarrow}{\cal D}\ \vert Z\vert$$for all $c\in (0,\infty),$ where $Z$ has the standard normal distribution. Result (ii) has a generalization to directly Riemann integrable functions.