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