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

TL;DR

This paper establishes a new log-type moment result for perpetuities, which are infinite sums of discounted i.i.d. variables, and applies it to analyze martingales in supercritical branching random walks, relaxing classical integrability conditions.

Contribution

It provides a minimal-condition log-moment result for perpetuities and links it to martingale limits in supercritical branching random walks, extending previous conditions for uniform integrability.

Findings

Derived a log-type moment result under minimal conditions.

Connected perpetuities to martingale limits in branching random walks.

Identified a necessary and sufficient condition for martingale uniform integrability.

Abstract

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 [Iksanov, A.M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Proc. Appl. 114, 27-50.], arises upon consideration of a size-biased version of the branching random walk originally introduced by Lyons in [Lyons, R.(1997). A simple path to Biggins' martingale convergence for branching random walk. In Athreya, K.B., Jagers, P. (eds.). Classical and Modern Branching Processes, IMA Volumes in Mathematics and its Applications, vol. 84, Springer, Berlin, 217-221.]. 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.