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

We consider a stochastic process $\xi(t)$, $t\ge 0$, $\xi(0)=0$, with independent stationary increments. Let $\eta(t,a)$ be a stochastic process with delay at the boundary of the half-interval $[-a,\infty)$, $a\ge0$, i.e., $\eta(t,a)=\xi(t)-a-\min\left\{-a;\ \inf_{s\le t}\xi(s)\right\}$. Under some restrictions on $\xi(1)$, we obtain asymptotic expansions for the Laplace–Stieltjes transforms of the normed random variable $\theta(a,b)=\inf\bigl\{t:\eta(t,a)\ge b\bigr\}$ as $b\to\infty$. The cases $\mathbb E\,\xi(1)=0$ and $\mathbb E\,\xi(1)<0$ are considered and the situations $a=\mathrm{const}$, $b\to\infty$; $a\to\infty$, $b\to\infty$; and $a\to\infty$, $b=\mathrm{const}$ are treated separately.