When Does a Branching Process Grow Like its Mean ?

The Kesten-Stigum Theorem is a fundamental criterion for the rate of growth of a supercritical branching process, showing that an L log L condition is decisive. In critical and subcritical cases, results of Kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations. Through a uniied approach, we give conceptual proofs of these theorems which are free of manipulation of generating functions. This approach also explains Yaglom's exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution. x1. Introduction. Consider a Galton-Watson branching process with each particle having probability p k of generating k children. Let L stand for a random variable with this progeny distribution. Let m := P k kp k be the mean number of children per particle and let Z n be the number of particles in the n th generation. The most basic and well-known fact about branching processes is that the extinction probability q := lim PZ n = 0] is equal to 1 if and only if m 1. It is also not hard to establish that in the case m > 1, 1 n log Z n ! log m almost surely on nonextinction, while in the case m 1, 1 n log PZ n > 0] ! log m: Finer questions may be asked: 1991 Mathematics Subject Classiication. Primary 60J80.