Asymptotic properties of supercritical branching processes in random environments

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 W_n=Z_n/\mathbb{E}[Zn|\xi ], 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).

Key words： Branching process; random environment; large deviation; moderate deviation; central limit theorem; moment; weighted moment; convergence rate

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