上临界多物种分支过程的小值概率

发布时间：2018-05-12 02:30

  本文选题：小值概率 + 上临界多物种分支过程　； 参考：《南京大学》2017年硕士论文

【摘要】：分支过程的小值概率是随机过程的极限性质研究的一个重要课题。而小值概率收敛到0的速率则和其分支机制是密切相关的,通常分为Schr(?)der case和B(?)ttcher case两种情况。在单物种的分支过程里,Schr(?)der case和B(?)ttcher case的分类由产生一个后代的概率决定。本文考虑的是上临界的多物种分支过程的小值概率问题。根据Jones(2004)[20]的结论,上临界的多物种分支过程的小值概率也是由分支机制所决定的。不过与单物种的分支过程比较起来,其Schr(?)der case和B(?)ttcher case的定义要复杂的多。从[20]关于Schr(?)der case和B(?)ttcher case的定义出发,我们对于两物种的分支过程得出更加具体的分类条件:可以从后代分布出发直接判断分支过程属于Schr(?)der case,还是B(?)ttcher case。我们从[20]定义里的判断矩阵A和最小后代矩阵B的谱半径与幂方收敛性出发,得到了两物种分支过程分类的充分条件,同时还得到了一些有趣的结论,如存在既不是Schr(?)der case也不是B(?)ttcher case的情况。此外,我们推广了文献Moters和Ortgiese[30]给出的概率方法,并给出上临界的多物种分支过程在B(?)ttcher case的小值概率上下界的一个证明。
[Abstract]:The probability of small value of branching process is an important subject in the study of the limit property of stochastic process. However, the rate of probability converging to zero is closely related to its branching mechanism, which is usually divided into two cases: Schr(?)der case and B(?)ttcher case. The classification of case and B(?)ttcher case in the branching process of a single species is determined by the probability of producing a progeny. In this paper, we consider the small probability problem of the upper critical multispecies branching process. According to the conclusion of Jones-2004 [20], the probability of small value of the upper critical multi-species branching process is also determined by the branching mechanism. But the definition of Schr(?)der case and B(?)ttcher case is much more complex than the branching process of a single species. Based on the definitions of Schr(?)der case and B(?)ttcher case in [20], we obtain more specific classification conditions for the branching processes of the two species: we can directly judge whether the branching processes belong to Schr(?)der case or B(?)ttcher case from the distribution of offspring. Starting from the spectral radius and power square convergence of the judgment matrix A and the minimal descendant matrix B in the definition of [20], we obtain sufficient conditions for the classification of the two species branching processes, and obtain some interesting conclusions at the same time. If there is neither Schr(?)der case nor B(?)ttcher case. In addition, we generalize the probabilistic methods given by Moters and Ortgiese [30], and give a proof of the upper lower bound of the lower probability of the upper critical multispecies branching process in B(?)ttcher case.
【学位授予单位】：南京大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.65
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本文编号：1876738