Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities

Abstract

Let $\mathcal{M}_{(n)},\quad n = 1, 2,...,$ be the supercritical branching random walk in which the family sizes may be infinite with positive probability. Assume that a natural martingale related to $\mathcal{M}_{(n)},$ converges almost surely and in the mean to a random variable $W$. For a large subclass of nonnegative and concave functions $f$ , we provide a criterion for the finiteness of $\mathbb{E}W f(W)$. The main assertions of the present paper generalize some results obtained recently in Kuhlbusch’s Ph.D. thesis as well as previously known results for the Galton-Watson processes. In the process of the proof, we study the existence of the $f$-moments of perpetuities.