隨機(jī)環(huán)境中多型分枝過程研究概述及一類鞅收斂

張影,彭雪蓮,王月嬌

(1. 長(zhǎng)沙理工大學(xué) 數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,湖南 長(zhǎng)沙,410000;2. 中南大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,湖南 長(zhǎng)沙,410083)

隨機(jī)環(huán)境中多型分枝過程研究概述及一類鞅收斂

張影1,彭雪蓮1,王月嬌2

(1. 長(zhǎng)沙理工大學(xué) 數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,湖南 長(zhǎng)沙,410000;2. 中南大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,湖南 長(zhǎng)沙,410083)

隨機(jī)環(huán)境;多型分枝過程;鞅

本文將對(duì)1971年至今的MBPRE的研究現(xiàn)狀進(jìn)行簡(jiǎn)要的概述,并構(gòu)造一個(gè)鞅過程,討論其極限隨機(jī)變量存在性。

1 MBPRE的研究情況概述

1.1 滅絕問題

本文將一些文獻(xiàn)的研究結(jié)論進(jìn)行符號(hào)上統(tǒng)一并做簡(jiǎn)要概述。首要問題就是研究其分類,不過在研究滅絕問題之后,分類問題隨之也會(huì)解決。Athreya、Karlin、E W Weissner、N Kaplan、D tanny都對(duì)MBPRE的滅絕問題進(jìn)行了研究,先來看他們所給出的滅絕條件。

N Kaplan[4]也討論了有關(guān)MBPRE的滅絕問題,在存在常數(shù)C,D＞0情況下有

1.2 滅絕時(shí)間的漸近性

1.3 上臨界極限問題

1.4 大數(shù)定律

1981年,D Tanny[5]在給出大數(shù)定律表達(dá)式前,先對(duì)界C及廣義的MBPRE進(jìn)行了定義,指出其大數(shù)定律形式:

1.5 拓展問題

2 一類鞅收斂

根據(jù)上述構(gòu)造的鞅是否可以研究MBPRE的中心極限定理問題,關(guān)于它的大偏差問題是否存在,矩的有限性及比率定理問題等可以繼續(xù)深入探討。

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

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

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[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.

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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

張影,1102573750@qq.com。

2017–01–20

國(guó)家自然科學(xué)基金(11571052,11171044);湖南省研究生科研創(chuàng)新項(xiàng)目(CX2016B417)。

(責(zé)任編校:劉剛毅)