The two-sided exit problem for a random walk on $\mathbb{Z}$ and having infinite variance II

TL;DR

This paper analyzes the asymptotic behavior of a random walk on the integers with infinite variance, focusing on the two-sided exit problem and ladder height processes under certain regularity conditions.

Contribution

It extends previous work by deriving asymptotic formulas for the ladder height renewal function and exit probabilities for oscillatory walks with infinite variance.

Findings

Asymptotic relation between renewal function and tail distribution established.

Probability estimates for exit on the upper side of an interval derived.

Asymptotic behavior of hitting probabilities conditioned on avoiding the negative half-line analyzed.

Abstract

Let $F$ be a distribution function on the integer lattice $\mathbb{Z}$ and $S=(S_n)$ the random walk with step distribution $F$. Suppose $S$ is oscillatory and denote by $U_{\rm a}(x)$ and $u_{\rm a}(x)$ the renewal function and sequence, respectively, of the strictly ascending ladder height process associated with $S$. Putting $A(x) =\int_0^x [1-F(t)-F(-t)] dt$, $H(x)=1-F(x)+F(-x)$ we suppose $$A(x)/\big(xH(x)\big) \to -\infty \quad (x\to\infty).$$ Under some additional regularity condition on the positive tail of $F$, we show that $$u_{\rm a}(x) \sim U_{\rm a}(x)[1-F(x)]/|A(x)|$$ as $x\to\infty$ and uniformly for $0\leq x\leq R\in \mathbb{Z}$, as $R\to\infty$ $$P [ S\; \mbox{leaves $[0,R]$ on its upper side}\, |\, S_0=x] \, \sim\, c^{-1}A(x)u_{\rm a}(x), $$ where $c= \sum_{n=1}^\infty P[S_{n}>S_0;\, S_k < S_0$ for $0<k<n]$ and the regularity condition is satisfied at least if $S$ is recurrent, $\limsup [1-F(x)]/F(-x)<1$, and $x[1-F(x)]/L(x) $ ($x\geq 1$) is bounded away from zero and infinity for some slowly varying function $L$. We also give some asymptotic estimates of the probability that $S$ visits $R$ before entering the negative half-line for asymptotically stable walks and obtain asymptotic behaviour of the probability that $R$ is ever hit by $S$ conditioned to avoid the negative half-line forever.