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

  • Yanqing Wang EMAIL logo and Quansheng Liu
Published/Copyright: November 11, 2021
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Abstract

This is a short survey about asymptotic properties of a supercritical branching process ( 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 ) , about the a.s. convergence, the non-degeneracy of its limit 𝑊, the convergence in L p for p 1 , and the boundedness of the harmonic moments E W n - a , a > 0 . We then present limit theorems and large deviation results on 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.

MSC 2010: 60F10; 60J80; 60K37

Award Identifier / Grant number: 2722021AJ014

Award Identifier / Grant number: 11731012

Funding statement: The work has been partially supported by the Fundamental Research Funds for the Central Universities (Grant No. 2722021AJ014) and the National Natural Science Foundation of China (Grant No. 11731012).

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Received: 2021-07-31
Revised: 2021-10-22
Accepted: 2021-10-22
Published Online: 2021-11-11
Published in Print: 2022-01-01

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