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Integro-local limit theorems for supercritical branching process in a random environment

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  • Struleva, M.A.
  • Prokopenko, E.I.

Abstract

Let Zn be a supercritical branching process in a random environment (BPRE). Under certain moment assumptions, we present the precise asymptotics for the “integro-local” probabilities P(logZn∈[x(n),x(n)+Δn)), where Δn→0 and x(n)→∞ as n→∞. In particular, this implies the large deviations tail asymptotics for P(logZn⩾x(n)) as n→∞. Like in previous research, we can see that, in the light-tail case, the main term in the large deviations asymptotics for the BPRE is provided by the associated random walk.

Suggested Citation

  • Struleva, M.A. & Prokopenko, E.I., 2022. "Integro-local limit theorems for supercritical branching process in a random environment," Statistics & Probability Letters, Elsevier, vol. 181(C).
    • Handle: RePEc:eee:stapro:v:181:y:2022:i:c:s0167715221001966
    DOI: 10.1016/j.spl.2021.109234
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    References listed on IDEAS

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    1. 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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