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 theL2point of view is given. IfZn(x) is the number of nth-generation particles to the left ofx, then under appropriate conditions for suitably chosenxn,Zn(xn)/Zn(+∞) converges inL2completely 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 intervalInin the age-dependent Crump–Mode–Jagers process is obtained.