Branching random walk in varying environments

Abstract

The branching random walk model is generalized towards generation-dependent displacement and reproduction distributions. Asymptotic theory of branching random walk in varying environments from the L 2 point of view is given. If Z n ( x ) is the number of nth-generation particles to the left of x , then under appropriate conditions for suitably chosen x n , Z n ( x n )/ Z n (+∞) converges in L 2 completely to a limiting distribution. Sufficient conditions for almost sure convergence are given. As a corollary an analogue of the central limit theorem for the proportion of particles of the nth generation in time interval I n in the age-dependent Crump–Mode–Jagers process is obtained.