The xlogx condition for general branching processes

Abstract

The xlogx condition is a fundamental criterion for the rate of growth of a general branching process, being equivalent to non-degeneracy of the limiting random variable. In this paper we adopt the ideas from Lyons, Pemantle and Peres (1995) to present a new proof of this well-known theorem. The idea is to compare the ordinary branching measure on the space of population trees with another measure, the size-biased measure.