2008/04/07 by Gerold Alsmeyer, Alsmeyer, Gerold, Alexander Iksanov +1
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.48550/arxiv.0804.0961
32 pages, submitted
arxiv created 2008/04/07 · openalex publication_date 2008/04/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.