On Lα-convergence (1≤α≤2) for a bisexual branching process with population-size dependent mating

Abstract

We investigate the L^α-convergence, 1≤α≤2, of the class of bisexual branching processes with population-size dependent mating, suitably normalized, to a finite limit W such that P(W>0)>0. Through different probabilistic approaches, we provide some necessary and sufficient conditions for such convergence. In particular we establish, by analogy with the classical Kesten and Stigum result for Bienaymé-Galton-Watson processes, a logarithmic criterion for L¹-convergence.