Convergence in Lp and its exponential rate for a branching process in a random environment

Chunmao Huang (Harbin Institute of Technology at Weihai)
Quansheng Liu (University Bretagne -Sud)

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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Pages: 1-22

Publication Date: November 3, 2014

DOI: 10.1214/EJP.v19-3388

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