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2014 Convergence in $L^p$ and its exponential rate for a branching process in a random environment
Chunmao Huang, Quansheng Liu
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Electron. J. Probab. 19: 1-22 (2014). DOI: 10.1214/EJP.v19-3388

Abstract

We consider a supercritical branching process $(Z_n)$ in a random environment $\xi$. Let $W$ be the limit of the normalized population size $W_n=Z_n/\mathbb{E}[Z_n|\xi]$. We first show a necessary and sufficient condition for the quenched $L^p$ ($p > 1$) convergence of $(W_n)$, which completes the known result for the annealed $L^p$ convergence. We then show that the convergence rate is exponential, and we find the maximal value of $\rho > 1$ such that $\rho^n(W-W_n)\rightarrow 0$ in $L^p$, in both quenched and annealed sense. Similar results are also shown for a branching process in a varying environment.

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Chunmao Huang. Quansheng Liu. "Convergence in $L^p$ and its exponential rate for a branching process in a random environment." Electron. J. Probab. 19 1 - 22, 2014. https://doi.org/10.1214/EJP.v19-3388

Information

Accepted: 3 November 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1307.60150
MathSciNet: MR3275856
Digital Object Identifier: 10.1214/EJP.v19-3388

Subjects:
Primary: 60K37
Secondary: 60J80

Keywords: branching process , exponential convergence rate , Lp convergence , moments , random environment , varying environment

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