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Generalized vector multiplicative cascades

Part of: Classical measure theory Stochastic processes Probability theory on algebraic and topological structures Markov processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Julien Barral
Julien Barral*
Affiliation:
INRIA Rocquencourt
*
∗ Postal address: Projet Fractales, INRIA Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France. Email address: julien.barral@inria.fr
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Abstract

We define the extension of the so-called ‘martingales in the branching random walk’ in R or C to some Banach algebras B of infinite dimension and give conditions for their convergence, almost surely and in the Lp norm. This abstract approach gives conditions for the simultaneous convergence of uncountable families of such martingales constructed simultaneously in C, the idea being to consider such a family as a function-valued martingale in a Banach algebra of functions. The approach is an alternative to those of Biggins (1989), (1992) and Barral (2000), and it applies to a class of families to which the previous approach did not. We also give a result on the continuity of these multiplicative processes. Our results extend to a varying environment version of the usual construction: instead of attaching i.i.d. copies of a given random vector to the nodes of the tree ∪n≥0N+n, the distribution of the vector depends on the node in the multiplicative cascade. In this context, when B=R and in the nonnegative case, we generalize the measure on the boundary of the tree usually related to the construction; then we evaluate the dimension of this nonstatistically self-similar measure. In the self-similar case, our convergence results make it possible to simultaneously define uncountable families of such measures, and then to estimate their dimension simultaneously.

Keywords

Self-similar cascadesbranching random walkrandom environmentrandom measuresHausdorff dimensionBanach-algebras-valued martingalesLp convergence

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G57: Random measures 28A80: Fractals
Secondary: 60G42: Martingales with discrete parameter 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case) 60F25: $L^p$-limit theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 4 , December 2001 , pp. 874 - 895
Copyright
Copyright © Applied Probability Trust 2001 

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