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Transition phenomena for ladder epochs of random walks with small negative drift

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Vitali Wachtel
Vitali Wachtel*
Affiliation:
University of Munich
*
∗ Postal address: Mathematical Institute, University of Munich, Theresienstrasse 39, D-80333, Munich, Germany. Email address: wachtel@math.lmu.de
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Abstract

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For a family of random walks {S(a)} satisfying E S1(a)=-a<0, we consider ladder epochs τ(a)=min {k≥1: Sk(a)<0}. We study the asymptotic behaviour, as a⇒0, of P (τ(a)>n) in the case when n=n(a)→∞. As a consequence, we also obtain the growth rates of the moments of τ(a).

Keywords

Random walkladder epochtransition phenomena

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G52: Stable processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 4 , December 2009 , pp. 1189 - 1214
Copyright
Copyright © Applied Probability Trust 2009 

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