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

Wachtel, V. I. and Denisov, D. E. and Korshunov, D. A. (2013) Tail asymptotics for the supercritical Galton-Watson process in the heavy-tailed case. Proceedings of the Steklov Institute of Mathematics, 282 (1). pp. 273-297. ISSN 0081-5438

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.