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

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 $U_{\mathrm{a}}(x)$ and $u_{\mathrm{a}}(x)$ 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 $$A(x)/(xH(x))\to -\infty (x\to \infty ).$$ Under some additional regularity condition on the positive tail of $F$, we show that $$u_{\mathrm{a}}(x)\sim U_{\mathrm{a}}(x)[1-F(x)]/|A(x)|$$ as $x\to \infty$ and uniformly for $0\le x\le R\in \mathbb{Z}$, 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, $\mathrm{limsup}[1-F(x)]/F(-x)<1$, and $x[1-F(x)]/L(x)$ ($x\ge 1$) 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.

41 pages; there were many minor mistakes of the the second version that are corrected in this one