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The overshoot of a random walk with negative drift. (English) Zbl 1108.60042
Summary: Let $$\{S_n,n\geq 0\}$$ be a random walk starting from 0 and drifting to $$-\infty$$, and let $$\tau(x)$$ be the first time when the random walk crosses a given level $$x\geq 0$$. Some asymptotics for the tail probability of the overshoot $$S_{\tau(x)}-x$$, associated with the event $$(\tau(x)<\infty)$$, are derived for the cases of heavy-tailed and light-tailed increments. In particular, the formulae obtained fulfill certain uniform requirements.

##### MSC:
 60G50 Sums of independent random variables; random walks
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