Weighted moments for Mandelbrot's martingales

Xingang Liang,Quansheng Liu

doi:10.1214/ecp.v20-4443

Abstract

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 $\ell$.

Highlights

Introduction and resultsWe consider a generalized Mandelbrot’s martingale (Yn) defined as sums of products of random weights indexed by nodes of a Galton-Watson tree
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 α (Y ), where α > 1 and is a positive function slowly varying at ∞

Summary

Introduction and results

We consider a generalized Mandelbrot’s martingale (Yn) defined as sums of products of random weights indexed by nodes of a Galton-Watson tree. For the Crump-Mode-Jirina process (where Ai ≤ 1 for all i), the equivalence between (a) and (c) was shown by Bingham and Doney (1975) when α > 1 is not an integer; when α > 1 is an integer, they showed that the equivalence remains true under the extra condition that the function (·) in the canonical representation form (1.3) of is positive and slowly varying at ∞. In the special case where Ai ≤ 1 for all i ≥ 1, Theorem 1.2 was first proved by Bingham and Doney (1975) in the context of Crump-Mode-Jirina processes, under the extra conditions that (·) is positive and slowly varying at ∞ and lim supn→∞ (bn)/ (an) < ∞ for all 1 < a < b < ∞; the last condition on the superior limit was removed by Iksanov and Rösler (2006). Our approach simplifies significantly the arguments in [1]; it leads to an uniform treatment for all α > 1, and enables us to remove the extra conditions on used in [1]

Auxiliary lemma

New proof of non-degeneration of Y