Tail asymptotics for the supercritical Galton–Watson process in the heavy-tailed case 1

As well known, for a supercritical Galton–Watson process Zn whose off-spring distribution has mean m> 1, the ratio Wn: = Zn/m n has a.s. limit, say W. We study tail behaviour of the distributions of Wn and W in the case where Z1 has heavy-tailed distribution, that is, EeλZ1 = ∞ for every λ> 0. We show how different types of distributions of Z1 lead to different asymptotic behaviour of the tail of Wn and W. We describe the most likel