On normalized multiplicative cascades under strong disorder

Abstract

Multiplicative cascades, under weak or strong disorder, refer to sequences of positive random measures n; ;n = 1;2;::: , parameterized by a positive disorder parameter , and defined on the Borel -field B of @T = f0;1;:::b 1g 1 for the product topology. The normalized cascade is defined by the corresponding sequence of random probability measures prob n; := Z 1 n; n; ;n = 1;2:::; normalized to a probability by the partition function Zn; . In this note, a recent result of Madaule [27, 2011] is used to explicitly construct a family of tree indexed probability measures prob1; for strong disorder parameters > c, almost surely defined on a common probability space. Moreover, viewingfprobn; : > cg 1 n=1 as a sequence of probability measure valued stochastic process leads to finite dimensional weak convergence in distribution to a probability measure valued processfprob1; : > cg. The limit process is