ON COMPACTNESS PROPERTIES OF THE EXIT POSITION OF A RANDOM WALK FROM AN INTERVAL

Abstract

We study the exit position $S_{T(r)} $ of a random walk $S_n$ from the interval $[-r, r]$, showing that the tightness of $\vert S_{T(r)}\vert / r$ is equivalent to a generalised kind of stochastic compactness of $S_n$ which we call $SC^\prime$.

This property is in turn equivalent to another kind of compactness property, which we call $SC^{\prime\prime}$, of the maximal sum ${S_n^\ast = \max_{1 \leq j \leq n}\vert S_j \vert}$.

The classes $SC^\prime$ and $SC^{\prime\prime}$, and a related class $SC_0$, which so far seem unexplored, are related to, but different from, the class of stochastically compact $S_n$ studied by Feller, and are similarly of interest in the study of the weak convergence properties of $S_n$ and $S_{T(r)}$.

We give equivalent characterisations of $SC'$ and $SC''$ in terms of the domination of $S_n$ and $S_n^*$ over their maximal increment, and also some analytic characterisations in terms of functionals of the underlying distribution. As a corollary we obtain an equivalence for the stochastic compactness of $\vert S_ {T(r)} \vert / r$.