Generalized vector multiplicative cascades

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.