Weighted moments of the limit of a branching process in a random environment

Xingang Liang (1) , Quansheng Liu (1)

1 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique

Let $(Z_n)$ be a supercritical branching process in an independent and identically distributed random environment $\zeta =(\zeta _{0},\zeta _{1},\ldots )$, and let $W$ be the limit of the normalized population size $Z_n/\mathbb{E}(Z_n|\zeta)$. We show a necessary and sufficient condition for the existence of weighted moments of $W$ of the form $\E W^{\alpha}\ell(W)$, where $\alpha\geq 1$, $\ell$ is a positive function slowly varying at $\infty$. In the Galton-Watson case, the results improve the corresponding ones of Bingham and Doney (1974) and Alsmeyer and Rösler (2004).

Domaines: Probabilités [math.PR]

Xingang Liang, Quansheng Liu. Weighted moments of the limit of a branching process in a random environment. Proceedings of Steklov institute, 2013, 282 (1), pp.127-145. ⟨hal-00908524⟩