Convergence in \(L^p\) and its exponential rate for a branching process in a random environment. (English) Zbl 1307.60150

Huang, Chunmao; Liu, Quansheng

Electron. J. Probab. 19, Paper No. 104, 22 p. (2014).

Summary: We consider a supercritical branching process \((Z_n)\) in a random environment \(\xi\). Let \(W\) be the limit of the normalized population size \(W_n=Z_n/\operatorname{E}[Z_n|\xi]\). 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.

MSC:

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

Keywords:

branching process; random environment; \(L^p\) convergence; exponential convergence rate