Electronic Journal of Probability

A Log-Type Moment Result for Perpetuities and Its Application to Martingales in Supercritical Branching Random Walks

Gerold Alsmeyer and Alex Iksanov

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Infinite sums of i.i.d. random variables discounted by a multiplicative random walk are called perpetuities and have been studied by many authors. The present paper provides a log-type moment result for such random variables under minimal conditions which is then utilized for the study of related moments of a.s. limits of certain martingales associated with the supercritical branching random walk. The connection arises upon consideration of a size-biased version of the branching random walk originally introduced by Lyons. As a by-product, necessary and sufficient conditions for uniform integrability of these martingales are provided in the most general situation which particularly means that the classical (LlogL)-condition is not always needed.

Article information

Electron. J. Probab., Volume 14 (2009), paper no. 10, 289-313.

Accepted: 29 January 2009
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G42: Martingales with discrete parameter
Secondary: 60K05: Renewal theory

branching random walk martingale moments perpetuity

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Alsmeyer, Gerold; Iksanov, Alex. A Log-Type Moment Result for Perpetuities and Its Application to Martingales in Supercritical Branching Random Walks. Electron. J. Probab. 14 (2009), paper no. 10, 289--313. doi:10.1214/EJP.v14-596. https://projecteuclid.org/euclid.ejp/1464819471

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