Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities - Iksanov, O. M. (Ukrains’kyi Matematychnyi Zhurnal, 2006)
Let M_(n),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 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 EW 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.