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Local asymptotics for the area under the random walk excursion

Part of: Stochastic processes Limit theorems

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.
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Abstract

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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

Information

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

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