Asymptotic Representations for Characteristics of Exit from an Interval for Stochastic Processes with Independent Increments

Given a homogeneous process $\xi(t)$ with independent increments, we consider the random variables $T=\inf\bigl\{t:\xi(t)\notin[-a,b]\bigr\}$ ($a\ge 0$, $b>0$) and $\xi(T)$, as well as $\theta$, the first passage time across the level $b$ by the process $\xi(t)-a-\min\Bigl\{-a,\ \inf\limits_{s\le t}\xi(s)\Bigr\}$. We find asymptotic expansions for the distribution $\xi(T)$ and for $\mathbb E T$ and $\mathbb E\theta$ as $b\to\infty$.