随机环境中两性分枝过程L^2-收敛的判别准则

A criterion law for L^2-convergence of a bisexual branching process in random environment

在随机环境中两性分枝过程L^1-收敛的对数判别准则的基础上,以条件均值增长率的上确界作为规范化因子,令{ε_k (ξ_n)}和{σ_k (ξ_n)}为非增长序列,当k≥k_0时,给出了W_n→WL^2的必要条件sum from k=0 to ∞ k^(-1)σ_k (ξ_n)<∞,同时求出了在一定条件下,当k≥1时,{W_n;n∈N}依L^2-收敛到非退化到的随机变量W的充分条件是sum from k=0 to ∞ k^(-1)σ_k (ξ_n)<∞和sum from k=0 to ∞ k^(-1)ε_k (ξ_n)<∞。

On the basis of the logarithmic criterion of L1-convergence of a bisexual branching process in random environment, taking the upper bound of conditional mean growth rate as the normalization factor, let {εκ（ξn）}和{σκ（ξn）} is nongrowth sequence, when k≥k0 , the necessary conditions∞∑κ=0k-1σκ（ξn）〈∞ of WnL2→W are put forward. Under certain conditions, when k ≥ 1 , the sufficient condition for {Wn;n∈N} to a nondegenerate random variable by L2-convergence is ∞∑κ=0k-1σκ（ξn）〈∞和∞∑κ=0k-1εκ（ξn）〈∞.