AS Convergence Rate and Lp-Convergence of Bisexual Branching Processes in a Random Environment and Varying Environment

Let (Z(n)) be a supercritical bisexual branching process in a random environment ?. We study the almost sure (a.s.) convergence rate of the submartingale W-n = Z(n)/I-n to its limit W, where (I-n) is an usually used norming sequence. We prove that under a moment condition of order p ? (1, 2), W -W-n = o(e(-na)) a.s. for some a > 0 that we find explicitly; assuming the logarithmic moment condition holds, we have W -W-n = o(n(-a)) a.s.. In order to obtain these results, we provide the L-p- convergence of (W-n); similar conclusions hold for a bisexual branching process in a varying environment.