Electronic Communications in Probability

Weighted moments for Mandelbrot's martingales

Xingang Liang and Quansheng Liu

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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 $\ell$.

Article information

Electron. Commun. Probab., Volume 20 (2015), paper no. 85, 12 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G18: Self-similar processes 60G42: Martingales with discrete parameter

weighted moments Mandelbrot’s martingale slowly varying function

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Liang, Xingang; Liu, Quansheng. Weighted moments for Mandelbrot's martingales. Electron. Commun. Probab. 20 (2015), paper no. 85, 12 pp. doi:10.1214/ECP.v20-4443. https://projecteuclid.org/euclid.ecp/1465321012

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