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A.s. convergence rate for a supercritical branching processes with immigration in a random environment

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  • Yingqiu Li
  • Xulan Huang

Abstract

Let (Zn) be a supercritical branching process with immigration (Yn) in a random environment ξ. We are interested in the almost sure (a.s.) convergence rate of the submartingale Wn=ZnΠn to its limit W where (Πn) is an usually used norming sequence. The result about convergence a.s. are as following. Under a moment condition of order p∈(1,2) and limn→∞ log m̂nn=0a.s. where, m̂n=EYn W−Wn=o(e−na) a.s. for some a > 0 that we find explicity; then assuming EW1 log W1α+1 0 we have W−Wn=o(n−α) a.s.; similar conclusions hold in a varying environment, but the condition limn→∞ log m̂nn=0a.s. will be replaced by ∑n=0∞anm̂nΠnmn

Suggested Citation

  • Yingqiu Li & Xulan Huang, 2022. "A.s. convergence rate for a supercritical branching processes with immigration in a random environment," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 51(3), pages 826-839, February.
    • Handle: RePEc:taf:lstaxx:v:51:y:2022:i:3:p:826-839
    DOI: 10.1080/03610926.2020.1756330
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    Cited by:

    1. Huang, Xulan & Li, Yingqiu & Xiang, Kainan, 2022. "Berry–Esseen bound for a supercritical branching processes with immigration in a random environment," Statistics & Probability Letters, Elsevier, vol. 190(C).

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