Weighted moments for Mandelbrot's martingales *
Résumé
Let (Yn) n≥0 be a Mandelbrot's martingale defined as sums of products of random weights indexed by nodes of a Galton-Watson tree, and let Y be its limit. We show a necessary and sufficient condition for the existence of weighted moments of Y of the forms EY^a l(Y)), where a > 1 and l is a positive function slowly varying at ∞. We also show a sufficient condition in the case of a = 1. Our results complete those of Alsmeyer and Kuhlbusch (2010) for weighted branching processes by removing their extra conditions on l.