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Asymptotics for the first passage times of Lévy processes and random walks. (English) Zbl 1264.60031
Summary: We study the exact asymptotics for the distribution of the first time \(\tau_{x}\) a Lévy process \(X_{t}\) crosses a fixed negative level \(-x\). We prove that \(\operatorname{P}\{\tau_{x} >t\} \sim V(x) \operatorname{P}\{X_{t}\geq 0\}/t\) as \(t\rightarrow \infty \) for a certain function \(V(x)\). Using known results for the large deviations of random walks, we obtain the asymptotic behavior of \(\operatorname{P}\{\tau _{x}>t\}\) explicitly in both light- and heavy-tailed cases.

MSC:
60G51 Processes with independent increments; Lévy processes
60G50 Sums of independent random variables; random walks
60K25 Queueing theory (aspects of probability theory)
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