Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-18T21:54:42.371Z Has data issue: false hasContentIssue false
  • English
  • Français

Local asymptotics for the area under the random walk excursion

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  26 July 2018

Elena Perfilev  and
Vitali Wachtel
Elena Perfilev*
Affiliation:
Universität Augsburg
Vitali Wachtel*
Affiliation:
Universität Augsburg
*
* 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 light-tailed increments. We determine the asymptotics for local probabilities for the area and prove a local central limit theorem for the duration of the excursion conditioned on the large values of its area.

Keywords

Random walkexcursionsubexponential distribution

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
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 2 , June 2018 , pp. 600 - 620
Copyright
Copyright © Applied Probability Trust 2018 

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

[1]Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing Office, Washington, D.C. Google Scholar
[2]Bateman, H. (1954). Tables of Integral Transforms, Vol. 1. McGraw-Hill, New York. Google Scholar
[3]Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York. Google Scholar
[4]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. Google Scholar
[5]Caravenna, F. and Chaumont, L. (2013). An invariance principle for random walk bridges conditioned to stay positive. Electron. J. Prob. 18, 60. Google Scholar
[6]Dobrushin, R. and Hryniv, O. (1996). Fluctuations of shapes of large areas under paths of random walks. Prob. Theory Relat. Fields 105, 423–458. Google Scholar
[7]Doney, R. A. (1983). A note on conditioned random walk. J. Appl. Prob. 20, 409–412. Google Scholar
[8]Duffy, K. R. and Meyn, S. P. (2014). Large deviation asymptotics for busy periods. Stoch. Systems 4, 300–319. Google Scholar
[9]Gel'fond, A. O. (1967). The Calculus of Finite Differences, 3rd edn. Nauka, Moscow (in Russian). Google Scholar
[10]Guillemin, F. and Pinchon, D. (1998). On the area swept under the occupation process of an M/M/1 queue in a busy period. Queueing Systems Theory Appl. 29, 383–398. Google Scholar
[11]Kearney, M. J. (2004). On a random area variable arising in discrete-time queues and compact directed percolation. J. Phys. A 37, 8421–8431. Google Scholar
[12]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. Google Scholar
[13]Sohier, J. (2010). A functional limit convergence towards Brownian excursion. Preprint. Available at https://arxiv.org/abs/1012.0118. Google Scholar
[14]Vatutin, V. A. and Wachtel, V. (2009). Local probabilities for random walks conditioned to stay positive. Prob. Theory Relat. Fields 143, 177–217. Google Scholar
[15]Vysotsky, V. (2010). On the probability that integrated random walks stay positive. Stoch. Process. Appl. 120, 1178–1193. Google Scholar