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Generalized vector multiplicative cascades

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

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 L p 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≥0 N + 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

Information

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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