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Tail asymptotics for the supercritical Galton-Watson process in the heavy-tailed case |
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| Authors: | V I Wachtel D E Denisov D A Korshunov |
| Institution: | 1. Mathematisches Institut, Ludwig-Maximilians-Universit?t München, Theresienstr. 39, D-80333, München, Germany 2. School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK 3. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, pr. Akademika Koptyuga 4, Novosibirsk, 630090, Russia |
| 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, $\mathbb{E}e^{\lambda {\rm Z}_1 } = \infty $ for every λ > 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. |
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