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A log-type moment result for perpetuities and its application to martingales in the supercritical branching random walk

By Gerold Alsmeyer and Alexander Iksanov


SUMMARY. 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, first observed by the second author in [14], arises upon consideration of a size-biased version of the branching random walk originally introduced by Lyons [25]. We also provide a necessary and sufficient condition for uniform integrability of these martingales in the most general situation which particularly means that the classical (LlogL)-condition is not always needed. 1 Introduction an

Year: 2009
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