Asymptotic Properties of a Supercritical Branching Process with Immigration in a Random Environment

Wang Yanqing () and Liu Quansheng ()
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Wang Yanqing: School of Statistics and Mathematics, Zhongnan University of Economics and Law, 430074 Wuhan, P. R. China
Liu Quansheng: LMBA, UMR 6205, Université Bretagne Sud, F-56000, Vannes, France

Stochastics and Quality Control, 2021, vol. 36, issue 2, 145-155

Abstract: This is a short survey about asymptotic properties of a supercritical branching process ( Z n ) (Z_{n}) with immigration in a stationary and ergodic or independent and identically distributed random environment. We first present basic properties of the fundamental submartingale ( W n ) (W_{n}) , about the a.s. convergence, the non-degeneracy of its limit 𝑊, the convergence in L p L^{p} for p ≥ 1 p\geq 1 , and the boundedness of the harmonic moments E ⁢ W n - a \mathbb{E}W_{n}^{-a} , a > 0 a>0 . We then present limit theorems and large deviation results on log ⁡ Z n \log Z_{n} , including the law of large numbers, large and moderate deviation principles, the central limit theorem with Berry–Esseen’s bound, and Cramér’s large deviation expansion. Some key ideas of the proofs are also presented.

Keywords: Branching Processes; Random Environments; Law of Large Numbers; Central Limit Theorem; Berry–Esseen Bound; Large Deviations; Cramér’s Expansion (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1515/eqc-2021-0030

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