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

Alex Iksanov,Gerold Alsmeyer

doi:10.1214/ejp.v14-596

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 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.

Highlights

Introduction and resultsThis article provides conditions for the finiteness of certain log-type moments for the limit of iterated i.i.d. random linear functions, called perpetuities
A similar result (Theorem 1.4) will be formulated for the a.s. limit of a well-known martingale associated with the branching random walk introduced in Subsection 1.2
The main achievement is that necessary and sufficient momenttype conditions are given in a full generality

Summary

Introduction and results

This article provides conditions for the finiteness of certain log-type moments for the limit of iterated i.i.d. random linear functions, called perpetuities. The main achievement is that necessary and sufficient momenttype conditions are given in a full generality The latter means that, for the first time, we have dispensed with condition (7) below in the case of perpetuities and with condition (A1) stated after Theorem 1.3 in the case of branching random walks. We will describe and exploit an interesting connection between these at first glance unrelated models which emerges when studying the weighted random tree associated with the branching random walk under the so-called size-biased measure.

Perpetuities

The branching random walk and its intrinsic martingales

Auxiliary results

Modified branching random walk

Connection with perpetuities

Two further auxiliary results