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

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<mark>Journal publication date</mark>10/2013
<mark>Journal</mark>Proceedings of the Steklov Institute of Mathematics
Issue number1
Volume282
Number of pages25
Pages (from-to)273-297
Publication StatusPublished
Early online date22/10/13
<mark>Original language</mark>English

Abstract

As is well known, for a supercritical Galton-Watson process Z (n) whose offspring distribution has mean m > 1, the ratio W (n) := Z (n) /m (n) has almost surely a limit, say W. We study the tail behaviour of the distributions of W (n) and W in the case where Z (1) has a heavy-tailed distribution, that is, for every lambda > 0. We show how different types of distributions of Z (1) lead to different asymptotic behaviour of the tail of W (n) and W. We describe the most likely way in which large values of the process occur.