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

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

Zbl 1227.60055
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