Journal of Applied Probability

Asymptotics for the first passage times of Lévy processes and random walks

Denis Denisov and Vsevolod Shneer

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Abstract

We study the exact asymptotics for the distribution of the first time, τx, a Lévy process Xt crosses a fixed negative level -x. We prove that ℙ{τx >t} ~V(x) ℙ{Xt≥0}/t as t→∞ for a certain function V(x). Using known results for the large deviations of random walks, we obtain asymptotics for ℙ{τx>t} explicitly in both light- and heavy-tailed cases.

Article information

Source
J. Appl. Probab. Volume 50, Number 1 (2013), 64-84.

Dates
First available in Project Euclid: 20 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.jap/1363784425

Digital Object Identifier
doi:10.1239/jap/1363784425

Mathematical Reviews number (MathSciNet)
MR3076773

Zentralblatt MATH identifier
1264.60031

Subjects
Primary: 60G50: Sums of independent random variables; random walks 60G51: Processes with independent increments; Lévy processes
Secondary: 60K25: Queueing theory [See also 68M20, 90B22]

Keywords
Lévy process random walk busy period first passage time subexponential distribution large deviation single-server queue

Citation

Denisov, Denis; Shneer, Vsevolod. Asymptotics for the first passage times of Lévy processes and random walks. J. Appl. Probab. 50 (2013), no. 1, 64--84. doi:10.1239/jap/1363784425. https://projecteuclid.org/euclid.jap/1363784425


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