Conceptual Proofs of L Log L Criteria for Mean Behavior of Branching Processes

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. We give conceptual proofs of these theorems based on comparisons of Galton-Watson measure to another measure on the space of trees. This approach also explains Yaglom's exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution.