On the stationary measures of critical branching processes

Abstract

Let Z n (n=0, l, ...) be an aperiodic critical Galton-Watson process and let σ2 be the (possibly infinite) variance of Z1. Let η k (k=1, 2, ...) denote the stationary measure of the process. Kesten, Ney and Spritzer proved in 1966 that η k →2/σ2 as k→∞ (*) under the additional assumption that EZ 1 2 log Z1<∞ (**) In the present paper, (*) is proved without the assumption (**). The proof uses complex function theory.