On Limit Theorems for the First Exit Time from a Strip for Stochastic Processes. I

We consider a stochastic process $\xi(t)$, $t\ge 0$, $\xi(0)=0$, with independent stationary increments. Let $T=T(a,b)=\inf\bigl\{t>0:\xi(t)\notin[-a,b)\bigr\}$, $a>0$, $b>0$. Under some restrictions on $\xi(1)$, we obtain asymptotic expansions as $a+b\to\infty$ for the Laplace–Stieltjes transforms of the suitably normed random variable $T$ with a fixed direction of exit. The cases $\mathbb E\,\xi(1)=0$ and $\mathbb E\,\xi(1)<0$ are considered and the situations $a\to\infty$ and $a=\mathrm{const}$ are separately treated. We also show how to pass from the obtained results to asymptotic expansions for probabilities.