Limit Theorems for Branching Processes in Random Environments

We present some limit theorems for branching processes in random environments, which can be found in [3], [4] or [5]. Let (Zn) be a branching process in a random environment. If the process is critical or subcritical, we show the exact decay rate of the survival probability P (Zn> 0) (as n!1), and nd a sequence (Cn) depending only on the environment, such that, conditional on Zn> 0, the normalized population Zn=Cn, converges in law to a non-trivial probability. If the process is supercritical, we obtain a necessary and sucient condition for the convergence in Lp of the natural martingale (Zn=Pn), where p> 1 is given and Pn is the conditional expected value of Zn given the environment. For example, in the critical case, we have the following analogues of laws of Yaglom and Kolmogorov. (We denote by E! the conditional expectation given the environment!.) Theorem [4,5]. Let (Zn) be a branching process in an i.i.d. random environment with probability generating functions (fn). Assume that E log f 0 1 (1) = 0, 0 < E[log f