正则化的连续时间马尔可夫分支过程的加权矩

Weighted Moments for the Limit of a Normalized Markov Branching Process

摘要: 设为一连续时间超临界马尔可夫分支过程，令$W$ 表示归一化种群数量$Z\left(t\right)/e^{\lambda t}$ 的极限，其中$e^{\lambda t}$ 为该分支过程的均值。设$l$ 为在无穷远处缓变的正函数。本文证明：对任意$a>1$ ，$E{W}^{\alpha }l\left(W\right)<\infty$ 当且仅当$E{Y}^{\alpha }l\left(Y\right)<\infty$ ，其中$Y$ 为子代数目。

Abstract: Let be a continuous-time supercritical Markov branching process, and let $W$ be the limit of the normalized population size $Z\left(t\right)/e^{\lambda t}$ , where $e^{\lambda t}$ is the mean of the branching process. Let $l$ be a positive function slowly varying at $\infty$ . In this paper, we prove that for $a>1$ , $E{W}^{\alpha }l\left(W\right)<\infty$ if and only if $E{Y}^{\alpha }l\left(Y\right)<\infty$ , where $Y$ is the number of offspring.