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

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 .