Asian Journal of Mathematics

Volume 23 (2019)

Number 6

Quenched weighted moments of a supercritical branching process in a random environment

Pages: 969 – 984



Yuejiao Wang (School of Mathematics and Statistics, Central South University, Changsha, China; and College of Mathematics & Computational Science, Hunan First Normal University, Changsha, China)

Yingqiu Li (College of Mathematics and Computing Sciences, Changsha University of Science and Technology, Changsha, China)

Quansheng Liu (Laboratoire de Mathématiques en Bretagne Atlantique, Université Bretagne Sud, Vannes, France)

Zaiming Liu (School of Mathematics and Statistics, Central South University, Changsha, China)


We consider a supercritical branching process $(Z_n)$ in an independent and identically distributed random environment $\xi = (\xi_n)$. Let $W$ be the limit of the natural martingale $W_n = Z_n / E_\xi Z_n , n \geq 0$, where $E_\xi$ denotes the conditional expectation given the environment $\xi$. We find a necessary and sufficient condition for the existence of quenched weighted moments of $W$ of the form $E_\xi W^{\alpha} l(W)$, where $\alpha \gt 1$ and $l$ is a positive function slowly varying at $\infty$. The same conclusion is also proved for the maximum of the martingale $W^{\ast} = \sup_{n \geq 1} W_n$ instead of the limit variable $W$. In the proof we first show an extended version of Doob’s inequality about weighted moments for nonnegative submartingales, which is of independent interest.


branching process, random environment, weighted moments, Doob’s inequality, slowly varying function

2010 Mathematics Subject Classification

60G42, 60J80

1fundingThe work has been partially supported by the National Natural Science Foundation of China (Grants no. 11731012, no. 11571052, and no. 11901186), the Guangdong Natural Science Foundation (Grant no. 2018A030313954), the Fundamental Research Funds for the Central Universities of Central South University (2015zzts012), and the Centre Henri Lebesgue (CHL, ANR-11-LABX-0020-01, France).

Received 20 December 2016

Accepted 16 November 2018

Published 3 August 2020