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

Let $S=(S_n)$ be an oscillatory random walk on the integer lattice $\mathbb{Z}$ with i.i.d. increments. Let $V_{{\rm d}}(x)$ 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 $0\leq x\leq R$. When $S$ is attracted to a stable process of index $0<\alpha \leq 2$ and there exists $\rho= \lim P[S_n>0]$, the sufficient condition obtained are also necessary for $(*)$ and fulfilled if and only if $(\alpha\vee 1)\rho =1$, and some asymptotic estimates of the probability on the left side of $(*)$ are given in case $(\alpha\vee 1)\rho \neq 1$.