Geodesic
On Limit Theorems for the First Exit Time from a Strip for Stochastic Processes. II
Matematičeskie trudy, Tome 2 (1999) no. 1, pp. 121-139 
V. I. Lotov; V. R. Khodzhibaev. On Limit Theorems for the First Exit Time from a Strip for Stochastic Processes. II. Matematičeskie trudy, Tome 2 (1999) no. 1, pp. 121-139. http://geodesic.mathdoc.fr/item/MT_1999_2_1_a3/
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     title = {On {Limit} {Theorems} for {the~First} {Exit} {Time} from {a~Strip} for {Stochastic} {Processes.~II}},
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     url = {http://geodesic.mathdoc.fr/item/MT_1999_2_1_a3/}
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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.