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

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  • Tanny, David

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))

Suggested Citation

  • Tanny, David, 1988. "A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means," Stochastic Processes and their Applications, Elsevier, vol. 28(1), pages 123-139, April.
    • Handle: RePEc:eee:spapps:v:28:y:1988:i:1:p:123-139
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    Cited by:

    1. Gao, Zhi-Qiang, 2021. "Exact convergence rate in the central limit theorem for a branching process in a random environment," Statistics & Probability Letters, Elsevier, vol. 178(C).
    2. Gao, Zhiqiang & Liu, Quansheng, 2016. "Exact convergence rates in central limit theorems for a branching random walk with a random environment in time," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2634-2664.
    3. Alsmeyer, Gerold & Gröttrup, Sören, 2016. "Branching within branching: A model for host–parasite co-evolution," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1839-1883.
    4. Grama, Ion & Liu, Quansheng & Miqueu, Eric, 2017. "Berry–Esseen’s bound and Cramér’s large deviation expansion for a supercritical branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1255-1281.
    5. Vincent Bansaye, 2019. "Ancestral Lineages and Limit Theorems for Branching Markov Chains in Varying Environment," Journal of Theoretical Probability, Springer, vol. 32(1), pages 249-281, March.
    6. Kuhlbusch, Dirk, 2004. "On weighted branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 113-144, January.
    7. Afanasyev, V. I., 2001. "On the maximum of a subcritical branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 93(1), pages 87-107, May.
    8. Huang, Chunmao & Liu, Quansheng, 2012. "Moments, moderate and large deviations for a branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 522-545.

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