Conceptual Proofs of $L$ Log $L$ Criteria for Mean Behavior of Branching Processes
Ann. Probab., 23 (3):
1125--1138 (July 1995)DOI: 10.1214/aop/1176988176
Abstract
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 Kol-
mogorov 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 an-
other 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.