The Galton-Watson process conditioned on the total progeny

Abstract

Let Z k denote the number in the k th generation of a Galton-Watson process initiated by one individual and let N be the total progeny, i.e., As n → ∞ the limiting behaviour of the process { Z k , 0 ≦ k ≦ n } conditioned on the event { N = n } is studied. The results obtained are of exactly the same form for the subcritical, critical and supercritical cases. This is in marked contrast to the analogous situation got by conditioning on non-extinction by the n th generation and letting n → ∞. In the latter case the limiting results differ in form for the critical and non-critical cases.