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 $m>1$, 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, $\mathbb E\,e^{\lambda Z_1}=\infty$ 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.