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2015 Weighted moments for Mandelbrot's martingales
Xingang Liang, Quansheng Liu
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Electron. Commun. Probab. 20: 1-12 (2015). DOI: 10.1214/ECP.v20-4443


Let $(Y_n)_{n\ge0}$ 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 Alsmeyer and Kuhlbusch (2010) for weighted branching processes by removing their extra conditions on $\ell$.


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Xingang Liang. Quansheng Liu. "Weighted moments for Mandelbrot's martingales." Electron. Commun. Probab. 20 1 - 12, 2015.


Accepted: 11 November 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 1335.60062
MathSciNet: MR3434202
Digital Object Identifier: 10.1214/ECP.v20-4443

Primary: 60J80
Secondary: 60G18 , 60G42

Keywords: Mandelbrot’s martingale , slowly varying function , weighted moments

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