Geometric rate of growth in population-size-dependent branching processes

We consider a branching-process model { Z n }, where the law of offspring distribution depends on the population size. We consider the case when the means m n ( m n is the mean of offspring distribution when the population size is equal to n ) tend to a limit m > 1 as n →∞. For a certain class of processes { Z n } necessary conditions for convergence in L 1 and L 2 and sufficient conditions for almost sure convergence and convergence in L 2 of W n = Z n /m n are given.