A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means

David Tanny

Stochastic Processes and their Applications, 1988, vol. 28, issue 1, 123-139

Abstract: It is known that, for a branching process in a random environment (BPRE) {Zn}[infinity]n=0 having conditional means {m([xi]n)}[infinity]n=0 where [xi]=([xi]0, [xi]1,...) is the environmental sequence, Zn/[Pi]n-1i=0 m([xi]i) converges almost surely to a random variable W. On the set where W is different from zero, the latter result implies that "the BPRE is growing like the product of its means"; however, it is possible for the BPRE to be supercritical and still have a degenerate limit W. In this paper, a sharp martingale comparison method is introduced which results in our obtaining a necessary and sufficient condition for W to be non-degenerate. When the environments are independent and identically distributed, this condition reduces to W is nondegenerate if and only if E((Z1log+Z1)/m([xi]0))

Keywords: branching; process; in; a; random; environment; comparison; varying; environment; process (search for similar items in EconPapers)
Date: 1988
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Citations: View citations in EconPapers (10)

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