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

Abstract

Let $(Z_n)$ be a supercritical branching process with immigration $(Y_n)$ in a random environment ξ. We are interested in the almost sure (a.s.) convergence rate of the submartingale $W_n=\frac{Z_n}{{\Pi }_n}$ to its limit $W$ where $({\Pi }_n)$ is an usually used norming sequence. The result about convergence a.s. are as following. Under a moment condition of order $p\in (1,2)$ and $\underset{n\to \infty }{\lim }\frac{\hspace{0.17em}\text{log}\hspace{0.17em}{\widehat{m}}_n}{n}=0a.s.$ where, ${\widehat{m}}_n=\mathbb{E}{Y}_n$ $W-W_n=o(e^{-na})$ a.s. for some a > 0 that we find explicity; then assuming $\mathbb{E}{W}_1\hspace{0.17em}\text{log}\hspace{0.17em}{W}_1^{\alpha +1}<\infty$ for some $\alpha >0$ we have $W-W_n=o(n^{-\alpha })$ a.s.; similar conclusions hold in a varying environment, but the condition $\underset{n\to \infty }{\lim }\frac{\hspace{0.17em}\text{log}\hspace{0.17em}{\widehat{m}}_n}{n}=0a.s.$ will be replaced by ${\sum }_{n=0}^{\infty }\frac{a_n{\widehat{m}}_n}{{\Pi }_n{m}_n}<\infty$ where $(a_n)$ is a positive sequence of real numbers.

KEYWORDS:

• Branching process
• immigration
• varying environment
• random environment
• submartingale
• convergence rate

Acknowledgements

The authors are grateful to anonymous referees and Professor Quansheng Liu for their very valuable comments and remarks, which significantly contributed to improving the quality of the paper.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 11571052, 11731012), the Hunan Provincial Natural Science Foundation of China (Grant No. 2018JJ2417), and the Open Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Grant No. 2018MMAEZD02).