Regular variation in the branching random walk

2006/04/20 by Aleksander Iksanov, Iksanov, Aleksander, Sergey Polotskiy +1
Computer Science · Decision Sciences · Mathematics · #60J80 #Bayesian Methods and Mixture Models #FOS: Mathematics #Primary: 60G42 #Probability (math.PR) #Probability and Risk Models #Secondary: 60E99 #Stochastic processes and statistical mechanics #math.PR #msc:60E99 #msc:60G42 #msc:60J80

paper · pdf · doi:10.48550/arxiv.math/0604439

submitted

arxiv created 2006/04/20 · openalex publication_date 2006/04/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \\mm_n, n=0,1,...\ be the supercritical branching random walk starting with one initial ancestor located at the origin of the real line. For n=0,1,... let W_n be the moment generating function of \mm_n normalized by its mean. Denote by AW_n any of the following random variables: maximal function, square function, L₁ and a.s. limit W, \su |W-W_n|, \su |W_n+1-W_n|. Under mild moment restrictions and the assumption that \rP\W₁>x\ regularly varies at ∞ it is proved that \rP\AW_n>x\ regularly varies at ∞ with the same exponent. All the proofs given are non-analytic in the sense that these do not use Laplace-Stieltjes transforms. The result on the tail behaviour of W is established in two distinct ways.

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