Weighted moments for Mandelbrot’s martingales. (English) Zbl 1335.60062

Liang, Xingang; Liu, Quansheng

Electron. Commun. Probab. 20, Paper No. 85, 12 p. (2015).

Summary: Let \((Y_n)_{n\geq0}\) 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 \(\mathbb{E}Y^{\alpha}\ell(Y)\), where \(\alpha>1\) and \(\ell\) is a positive function slowly varying at \(\infty\). We also show a sufficient condition in the case of \(\alpha=1\). Our results complete those of G. Alsmeyer and D. Kuhlbusch [Münster J. Math. 3, No. 1, 163–212 (2010; Zbl 1227.60055)] for weighted branching processes by removing their extra conditions on \(\ell\).

MSC:

60G42	Martingales with discrete parameter
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60G18	Self-similar stochastic processes
26A12	Rate of growth of functions, orders of infinity, slowly varying functions

Keywords:

Mandelbrot martingale; weighted moments; slowly varying function

Citations:

Zbl 1227.60055