Electronic Journal of Probability

Convergence in $L^p$ and its exponential rate for a branching process in a random environment

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.

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 104, 22 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065746

Digital Object Identifier
doi:10.1214/EJP.v19-3388

Mathematical Reviews number (MathSciNet)
MR3275856

Zentralblatt MATH identifier
1307.60150

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Rights

Citation

Huang, Chunmao; Liu, Quansheng. Convergence in $L^p$ and its exponential rate for a branching process in a random environment. Electron. J. Probab. 19 (2014), paper no. 104, 22 pp. doi:10.1214/EJP.v19-3388. https://projecteuclid.org/euclid.ejp/1465065746

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