The two-sided exit problem for a random walk on $\mathbb{Z}$ with infinite variance I

Abstract

Let $S=(S_n)$ be an oscillatory random walk on the integer lattice $\mathbb{Z}$ with i.i.d. increments. Let $V_{\mathrm{d}}(x)$ be the renewal function of the strictly descending ladder height process for $S$. We obtain several sufficient conditions -- given in terms of the distribution function of the increment $S_1-S_0$ -- so that as $R\to \infty$ $$ (*) \quad P [ S\; \mbox{leaves $[0,R]$ on its upper side}\, |\, S_0=x] \, \sim\, V_{\rm d}(x)/V_{\rm d}(R)$$ uniformly for $0\le x\le R$. When $S$ is attracted to a stable process of index $0<\alpha \le 2$ and there exists $\rho =\lim P[S_n>0]$, the sufficient condition obtained are also necessary for $(\ast )$ and fulfilled if and only if $(\alpha \vee 1)\rho =1$, and some asymptotic estimates of the probability on the left side of $(\ast )$ are given in case $(\alpha \vee 1)\rho \ne 1$.

26 pages, Several minor errors found in the preceding version are corrected

References in corpus (1)

• Estimates of Potential functions of random walks on $Z$ with zero mean and infinite variance and their applications