Some Limit Theorems For Subcritical Branching Processes In Random Environment

Let ζ be a random environment,of which ζ:={ζ_n} = {ζ_n(ω):n∈N} is a stationary and ergodic process, {Z_n,n≥ 0} is a branching process in the random environment ζ ,The realization of the sequence of ζ:={ζ_n} determines a sequence of probalitity generating functions {f_n}(n≥0) of the branching process {Z_n,n≥0} in the random environment ζA branching process {Z_n,n≥0} in the random environment ζ (BPRE)isa family of time-inhomogeneous branching processes:given the environment sequence ζ,the process {Z_n,n≥0} acts as a Galton-Watsonprocess in varying environments with probalitily generating functions {f_n}(n ≥ 0).By definition,where conditioned on ζ ,(X_n,i,i≥1) are inter-valued random variables...