对于临界的随机环境中的分枝过程, 现有文献中研究了鞅的几乎处处收敛以及当相关联的随机游动二阶矩存在时鞅的条件L^1收敛的充分条件, 而本文考虑当二阶矩不存在即Cauchy 条件下临界的随机环境中的分枝过程, 得到在对原测度进行测度变换后, 与之相关联的鞅W_n=Z_n/e^{S_n}能够条件L^1收敛到一个非退化的随机变量的充分条件, 以及在什么条件下此鞅的条件L^1 极限退化.

For critical branching processes in random environment, the almost everywhere convergence of the martingales and the sufficient condition for the conditional L^1-convergence under the condition that the second moment of associated random walk exists have been studied. In this paper, we consider the second moment does not exist, that is, the critical branching process in the random environment under Cauchy condition. We give the sufficient condition that after changing of measure, the martingale W_n=Z_n/e^{S_n}  conditional L^1-converge to a non-degenerate random variable. Also, we give the condition that the conditional L^1 limit of this martingale degenerates.