Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-06-02T09:22:31.262Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotics for the First Passage Times of Lévy Processes and Random Walks

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  30 January 2018

Denis Denisov  and
Vsevolod Shneer
Denis Denisov*
Affiliation:
Cardiff University
Vsevolod Shneer*
Affiliation:
Heriot-Watt University
*
∗ Postal address: School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK.
∗∗ Postal address: Department of AMS, Heriot-Watt University, Edinburgh, EH14 4AS, UK. Email address: v.shneer@hw.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Lévy processrandom walkbusy periodfirst passage timesubexponential distributionlarge deviationsingle-server queue

MSC classification

Primary: 60G50: Sums of independent random variables; random walks 60G51: Processes with independent increments; L'evy processes
Secondary: 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 64 - 84
Copyright
Copyright © Applied Probability Trust 2013 

References

Abate, J. and Whitt, W. (1997). Asymptotics for M/GI/1 low-priority waiting-time tail probabilities. Queueing Systems 25, 173–233.CrossRefGoogle Scholar
Asmussen, S., Klüppelberg, C. and Sigman, K. (1999). Sampling at sub-exponential times, with queueing applications. Stoch. Process. Appl. 79, 265–286.CrossRefGoogle Scholar
Baltrūnas, A., Daley, D. J. and Klüppelberg, C. (2004). Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times. Stoch. Process. Appl. 111, 237–258.Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bertoin, J. and Doney, R. A. (1996). Some asymptotic results for transient random walks. Adv. Appl. Prob. 28, 207–226.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
Borovkov, A. A. (1962). New limit theorems in boundary problems for sums of independent terms. Siberian Math. J. 3, 645–694.Google Scholar
Borovkov, A. A. (2004). On the asymptotics of distributions of first-passage times. I. Math. Notes 75, 23–37.Google Scholar
Borovkov, A. A. (2004). On the asymptotics of distributions of first-passage times. II. Math. Notes 75, 322–330.Google Scholar
Cohen, J. W. (1982). The Single Server Queue, 2nd edn. North-Holland, Amsterdam.Google Scholar
De Meyer, A. and Teugels, J. L. (1980). On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1. J. Appl. Prob. 17, 802–813.Google Scholar
Denisov, D., Dieker, A. B. and Shneer, V. (2008). Large deviations for random walks under subexponentiality: the big-jump domain. Ann. Prob. 36, 1946–1991.Google Scholar
Doney, R. A. (1989). On the asymptotic behaviour of first passage times for transient random walk. Probab. Theory Relat. Fields 81, 239–246.CrossRefGoogle Scholar
Doney, R. A. and Maller, R. A. (2004). Moments of passage times for Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 40, 279–297.Google Scholar
Embrechts, P. and Hawkes, J. (1982). A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory. J. Austral. Math. Soc. Ser. A 32, 412–422.Google Scholar
Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth 49, 335–347.Google Scholar
Foss, S. and Korshunov, D. (2000). Sampling at a random time with a heavy-tailed distribution. Markov Process. Relat. Fields 6, 543–568.Google Scholar
Jelenković, P. R. and Momčilović, P. (2004). Large deviations of square root insensitive random sums. Math. Operat. Res. 29, 398–406.CrossRefGoogle Scholar
Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.Google Scholar
Klüppelberg, C. (1989). Subexponential distributions and characterizations of related classes. Prob. Theory Relat. Fields 82, 259–269.CrossRefGoogle Scholar
Kyprianou, A. E. and Palmowski, Z. (2006). Quasi-stationary distributions for Lévy processes. Bernoulli 12, 571–581.Google Scholar
Mogul'skii, A. A. (2006). Large deviations of the first passage time for a random walk with semiexponentially distributed Jumps. Siberian Math. J. 47, 1084–1101.CrossRefGoogle Scholar
Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Prob. 41, 407–424.Google Scholar
Palmowski, Z. and Rolski, T. (2006). On the exact asymptotics of the busy period in GI/G/1 queues. Adv. Appl. Prob. 38, 792–803.Google Scholar
Petrov, V. V. (1965). On the probabilities of large deviations for sums of independent random variables. Theory Prob. Appl. 10, 287–298.Google Scholar
Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.Google Scholar
Rogozin, B. A. (1966). Distribution of certain functionals related to boundary value problems for processes with independent increments. Theory Prob. Appl. 11, 656–670.CrossRefGoogle Scholar
Rozovskiı˘, L. V.. (1993). Probabilities of large deviations on the whole axis. Theory Prob. Appl. 38, 53–79.CrossRefGoogle Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Zwart, A. P. (2001). Tail asymptotics for the busy period in the GI/G/1 queue. Math. Operat. Res. 26, 485–493.Google Scholar