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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.
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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 

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