Weighted moments for Mandelbrot's martingales *

Xingang Liang (1) , Quansheng Liu (2)

1 Beijing Technology and Business University
2 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique

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.

Mots clés: weighted moments; Mandelbrot's martingale; slowly varying function

Domaines: Probabilités [math.PR]

Xingang Liang, Quansheng Liu. Weighted moments for Mandelbrot's martingales *. Electronic Communications in Probability, 2015, 20 (85), pp.1-12. ⟨10.1214/ECP.v20-4443⟩. ⟨hal-01267125⟩