Abstract
For a supercritical branching process, with immigration
in an independent and identically distributed random environment
. Let W be the limit of the submartingale
, where
is the usually used normalization sequence. The necessary and sufficient condition for the existence of quenched weighted moments of W of the form
is obtained in this article, where
and l is a positive function slowly varying at
. The same conclusion holds also for the maximum variable
.
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.