Geodesic
On Limit Theorems for the First Exit Time from a Strip for Stochastic Processes. I
Matematičeskie trudy, Tome 1 (1998) no. 2, pp. 111-134 
V. I. Lotov; V. R. Khodzhibaev. On Limit Theorems for the First Exit Time from a Strip for Stochastic Processes. I. Matematičeskie trudy, Tome 1 (1998) no. 2, pp. 111-134. http://geodesic.mathdoc.fr/item/MT_1998_1_2_a4/
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     author = {V. I. Lotov and V. R. Khodzhibaev},
     title = {On {Limit} {Theorems} for {the~First} {Exit} {Time} from {a~Strip} for {Stochastic} {Processes.~I}},
     journal = {Matemati\v{c}eskie trudy},
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     url = {http://geodesic.mathdoc.fr/item/MT_1998_1_2_a4/}
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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.