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

Yingqiu Li and Xulan Huang

Communications in Statistics - Theory and Methods, 2022, vol. 51, issue 3, 826-839

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

Date: 2022
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DOI: 10.1080/03610926.2020.1756330

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