Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T04:34:27.206Z Has data issue: false hasContentIssue false
  • English
  • Français

Tail asymptotics for the area under the excursion of a random walk with heavy-tailed increments

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  25 February 2021

Denis Denisov Open the ORCID record for Denis Denisov [Opens in a new window] ,
Elena Perfilev  and
Vitali Wachtel
Denis Denisov*
Affiliation:
University of Manchester
Elena Perfilev*
Affiliation:
Universität Augsburg
Vitali Wachtel*
Affiliation:
Universität Augsburg
*
*Postal address: Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK. Email address: [email protected]
**Postal address: Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany.
**Postal address: Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and heavy-tailed increments. We determine the asymptotics for tail probabilities for the area.

Keywords

Random walksubexponential distributionheavy-tailed distributionintegrated random walk

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60F17: Functional limit theorems; invariance principles
Type
Research Papers
Information
Journal of Applied Probability , Volume 58 , Issue 1 , March 2021 , pp. 217 - 237
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bateman, H. (1954). Tables of Integral Transforms, Vol. 1. McGraw-Hill, New York.Google Scholar
Borovkov, A. A., Boxma, O. J. and Palmowski, Z. (2003). On the integral of the workload process of the single server queue. J. Appl. Prob. 40, 200–225.CrossRefGoogle Scholar
Carmona, P. and Pétrélis, N. (2019). A shape theorem for the scaling limit of the IPDSAW at criticality. Ann. Appl. Prob. 29, 875–930.CrossRefGoogle Scholar
Denisov, D. (2005). A note on the asymptotics for the maximum on a random time interval of a random walk. Markov Process. Relat. Fields 11, 165–169.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.CrossRefGoogle Scholar
Denisov, D., Kolb, M. and Wachtel, V. (2015). Local asymptotics for the area of random walk excursions, J. London Math. Soc. 91, 495–513.CrossRefGoogle Scholar
Denisov, D. and Shneer, V. (2013). Asymptotics for the first passage times of Lévy processes and random walks. J. Appl. Prob. 50, 64–84.CrossRefGoogle Scholar
Dobrushin, R. and Hryniv, O. (1996). Fluctuations of shapes of large areas under paths of random walks. Prob. Theory Relat. Fields 105, 423–458.CrossRefGoogle Scholar
Doney, R. A. (1989). On the asymptotic behaviour of first passage times for transient random walks. Prob. Theory Relat. Fields 81, 239–246.CrossRefGoogle Scholar
Duffy, K. and Meyn, S. (2014). Large deviation asymptotics for busy periods. Stoch. Syst. 4, 300–319.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley, New York.Google Scholar
Foss, S., Palmowski, Z. and Zachary, S. (2005). The probability of exceeding a high boundary on a random time interval for a heavy-tailed random walk. Ann. Appl. Prob. 15, 1936–1957.CrossRefGoogle Scholar
Kulik, R. and Palmowski, Z. (2011). Tail behaviour of the area under a random process, with applications to queueing systems, insurance and percolations. Queueing Systems 68, 275–284.CrossRefGoogle Scholar
Perfilev, A. and Wachtel, V. (2018). Local asymptotics for the area under the random walk excursion. Adv. Appl. Prob. 50, 600–620.CrossRefGoogle Scholar
Rozovskii, L. V. (1993). Probabilities of large deviations on the whole axis. Theory Prob. Appl. 38, 53–79.CrossRefGoogle Scholar
Takacs, L. (1991). A Bernoulli excursion and its various applications. Adv. Appl. Prob. 23, 557–585.CrossRefGoogle Scholar
Takacs, L. (1992). On the total heights of random rooted trees. J. Appl. Prob. 29, 543–556.CrossRefGoogle Scholar
Takacs, L. (1994). On the total heights of random rooted binary trees. J. Combinatorial Theory B 61, 155–166.CrossRefGoogle Scholar