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

Original · EN

As well known, for a supercritical Galton-Watson process Zₙ whose offspring distribution has mean m>1, the ratio Wₙ:=Zₙ/mⁿ 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, eλᶻ¹=∞ 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 likely way how large values of the process occur.