随机环境中多型分枝过程研究概述及一类鞅收敛

张影,彭雪莲,王月娇

(1. 长沙理工大学 数学与计算科学学院,湖南 长沙,410000;2. 中南大学 数学与统计学院,湖南 长沙,410083)

随机环境中多型分枝过程研究概述及一类鞅收敛

张影1,彭雪莲1,王月娇2

(1. 长沙理工大学 数学与计算科学学院,湖南 长沙,410000;2. 中南大学 数学与统计学院,湖南 长沙,410083)

随机环境;多型分枝过程;鞅

本文将对1971年至今的MBPRE的研究现状进行简要的概述,并构造一个鞅过程,讨论其极限随机变量存在性。

1 MBPRE的研究情况概述

1.1 灭绝问题

本文将一些文献的研究结论进行符号上统一并做简要概述。首要问题就是研究其分类,不过在研究灭绝问题之后,分类问题随之也会解决。Athreya、Karlin、E W Weissner、N Kaplan、D tanny都对MBPRE的灭绝问题进行了研究,先来看他们所给出的灭绝条件。

N Kaplan[4]也讨论了有关MBPRE的灭绝问题,在存在常数C,D＞0情况下有

1.2 灭绝时间的渐近性

1.3 上临界极限问题

1.4 大数定律

1981年,D Tanny[5]在给出大数定律表达式前,先对界C及广义的MBPRE进行了定义,指出其大数定律形式:

1.5 拓展问题

2 一类鞅收敛

根据上述构造的鞅是否可以研究MBPRE的中心极限定理问题,关于它的大偏差问题是否存在,矩的有限性及比率定理问题等可以继续深入探讨。

[1] Athreya K B,Ney P E. Branching processes [M]. Berlin:Springer,1972:181–219.

[2] Athreya K B,Karlin S. on branching process with random environments:I:E-xtinction probabilities [J]. Adv Appl Probab,1971,42(5):1 499–1 520.

[3] Weissner E W. Multitype branching processes in random environments [J]. Journal of Applied Probability,1971,8(1):17–31.

[4] Kaplan N. Some results about multidimensional branching Processes with Random Environments [J]. The annuls of Probability,1974,2(3):441–455.

[5] Tanny D. On Multitype branching processes in a random environment [J]. Advances in Applied Probability,1981,13(3):464–497.

[6] Dyakonova E E. Asymptotic behaviour of probobility of non-extincton for a multi-type branching process in random environment [J]. Discrete Math Appl,1999,9(2):119–136.

[7] Dyakonova E E. Multitype branching processes in a Random Environment [J]. J M Sciences,2002,111(3):3 537–3 540.

[8] Harry Cohn. On the growth of the multitype supercritical branching process in a random environment [J]. The Ann Probability,1989,17(3):1 118–1 123.

[9] Vatutin V A.Polling system and multitype branching processes in random environment counted by random characteristics[J]. Math P R,2009,10(6):156–194.

[10] Mode C D,Raj T,Sleeman C K. Simulating the emergence and survival of mutations using a self regulating multitype branching processes [J]. Probability and Statistics,2011:223–262.

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Summary of research and a class of martingale convergence for multi-type branching process in random environments

Zhang Ying1,Peng Xuelian1,Wang Yuejiao2
(1. School of Mathematics,Changsha University of Science and Technology,Changsha 410000,China;2. School of Mathematics and Statistics of CSU,Changsha 410083,China)

random environment;multi-type branching process;martingale

O 211.65

A

1672–6146(2017)04–0008–04

doi∶ 10.3969/j.issn.1672–6146.2017.04.003

张影,1102573750@qq.com。

2017–01–20

国家自然科学基金(11571052,11171044);湖南省研究生科研创新项目(CX2016B417)。

(责任编校:刘刚毅)