Quenched weighted moments for a branching process with immigration in a random environment

Abstract

For a supercritical branching process, ${(Z_n)}_{n\ge 0}$ with immigration ${(Y_n)}_{n\ge 0}$ in an independent and identically distributed random environment $\xi ={({\xi }_n)}_{n\ge 0}$. Let W be the limit of the submartingale $W_n=Z_n/{\Pi }_n,n\ge 0$, where ${({\Pi }_n)}_{n\ge 0}$ is the usually used normalization sequence. The necessary and sufficient condition for the existence of quenched weighted moments of W of the form ${\mathbb{E}}_{\xi }\left[W^{\alpha }l(W)\right]$ is obtained in this article, where $\alpha >1,$ and l is a positive function slowly varying at $\infty$. The same conclusion holds also for the maximum variable $W^{*}=\underset{n\ge 1}{\sup }{W}_n$.

Keywords:

• Branching process with immigration
• random environment
• submartingale
• weighted moments

2020 MSC:

• 60J80
• 60F05

Acknowledgments

The author is grateful to the anonymous referees for very valuable comments and remarks which significantly contributed to improving the quality of the article. I would like to thank Professors Kainan Xiang and Yingqiu Li for their very useful and stimulating discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Nos. 12171410, 11871032), the Hu Xiang Gao Ceng Ci Ren Cai Ju Jiao Gong Cheng-Chuang Xin Ren Cai (No. 2019RS1057), the Hunan Provincial Natural Science Foundation of China (No. 2018JJ2417), the Graduate Innovation Project of Xiangtan University (No. XDCX2022Y058), and the Hunan Provincial Innovation Foundation For Postgraduate (No. CX20220637).