  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 $(\alpha\vee
1)\rho \neq 1$.
