Tail asymptotics for the supercritical Galton-Watson process in the heavy-tailed case

2013/03/10 by Denis Denisov, Denisov, Denis, Dmitry Korshunov +3
Economics, Econometrics and Finance · Mathematics · #60F10 #60G70 #60J80 #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60F10 #msc:60G70 #msc:60J80

paper · pdf · doi:10.48550/arxiv.1303.2306

arxiv created 2013/03/10 · openalex publication_date 2013/03/10 · arxiv updated 2013/03/12 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

As well known, for a supercritical Galton-Watson process Z_n whose offspring distribution has mean m>1, the ratio W_n:=Z_n/mⁿ has a.s. limit, say W. We study tail behaviour of the distributions of W_n and W in the case where Z₁ has heavy-tailed distribution, that is, \E e^λZ₁=∞ for every λ>0. We show how different types of distributions of Z₁ lead to different asymptotic behaviour of the tail of W_n and W. We describe the most likely way how large values of the process occur.

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