论文标题
关键的分支过程在不利的随机环境中演变
Critical branching processes evolving in an unfavorable random environment
论文作者
论文摘要
令$ \ left \ {z_ {n}，n = 0,1,2，... \ right \} $是随机环境中的关键分支过程，让$ \ left \ left \ {s_ {n}，n = 0,1,2，... \ right \} $是与随机步行相关的。众所周知，如果此随机步行的增量属于（不居中）稳定定律吸引的领域，则存在一个序列$ a_ {1}，a_ {2}，... $ $在无限慢慢变化，以至于条件分布 \frac{S_{n}}{a_{n}}\leq x\Big|Z_{n}>0\right) ,\quad x\in (-\infty ,+\infty ), \end{equation*}% weakly converges, as $n\rightarrow \infty $ to the distribution of a strictly positive and proper random variable.在本文中，我们用描述了概率的渐近行为\ begin {equation*} \ mathbf {p} \ left（s_ {n} \ leq或\ infty $以$φ（n）= o（a_ {n}）$的方式。
Let $\left\{ Z_{n},n=0,1,2,...\right\} $ be a critical branching process in random environment and let $\left\{ S_{n},n=0,1,2,...\right\} $ be its associated random walk. It is known that if the increments of this random walk belong (without centering) to the domain of attraction of a stable law, then there exists a sequence $a_{1},a_{2},...,$ slowly varying at infinity such that the conditional distributions \begin{equation*} \mathbf{P}\left( \frac{S_{n}}{a_{n}}\leq x\Big|Z_{n}>0\right) ,\quad x\in (-\infty ,+\infty ), \end{equation*}% weakly converges, as $n\rightarrow \infty $ to the distribution of a strictly positive and proper random variable. In this paper we supplement this result with a description of the asymptotic behavior of the probability \begin{equation*} \mathbf{P}\left( S_{n}\leq φ(n);Z_{n}>0\right) , \end{equation*}% if $φ(n)\rightarrow \infty $ \ as $n\rightarrow \infty $ in such a way that $φ(n)=o(a_{n})$.
