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

As is well known, for a supercritical Galton-Watson process Z n whose offspring distribution has mean m {\textgreater} 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, ????{e??Z}1=???{\textbackslash}mathbb\{E\}e{\textasciicircum}\{{\textbackslash}lambda \{{\textbackslash}rm Z\}\_1 \} = {\textbackslash}infty for every ?? {\textgreater} 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.