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 / $\mathbb{E}$ [Z n |ξ], 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 ).