Asymptotic properties of supercritical branching processes in random environments

doi:10.1007/s11464-014-0397-z

Front. Math. China

2014, Vol. 9

Issue (4) : 737-751 https://doi.org/10.1007/s11464-014-0397-z

SURVEY ARTICLE

Asymptotic properties of supercritical branching processes in random environments

Yingqiu LI¹,Quansheng LIU^1,^2,^*(

),Zhiqiang GAO³,Hesong WANG¹

1. College of Mathematics and Computing Sciences, Changsha University of Science and Technology, Changsha 410004, China
2. Université de Bretagne-Sud, UMR 6205, LMBA, F-56000 Vannes, France
3. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

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Abstract

We consider a supercritical branching process (Z_n) in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the natural martingale Wn=Zn/E[Zn|ξ], the convergence rates of W-W_n(by considering the convergence in law with a suitable norming, the almost sure convergence, the convergence in L^P, and the convergence in probability), and limit theorems (such as central limit theorems, moderate and large deviations principles) on (log Z_n).

Keywords Branching process random environment large deviation moderate deviation central limit theorem moment weighted moment convergence rate

Corresponding Author(s): Quansheng LIU

Issue Date: 26 August 2014

Cite this article:

Yingqiu LI,Quansheng LIU,Zhiqiang GAO, et al. Asymptotic properties of supercritical branching processes in random environments[J]. Front. Math. China, 2014, 9(4): 737-751.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-014-0397-z
https://academic.hep.com.cn/fmc/EN/Y2014/V9/I4/737

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