On Normalized Multiplicative Cascades under Strong Disorder

TL;DR

This paper constructs and analyzes the limiting behavior of normalized multiplicative cascades under strong disorder, revealing convergence to a process built from derivative martingales and decorated Poisson processes.

Contribution

It explicitly constructs the limiting probability measures for strong disorder cascades and establishes their convergence properties, extending previous results with new probabilistic constructions.

Findings

Explicit construction of limit measures for strong disorder cascades

Weak convergence of finite-dimensional distributions to the limit process

Connection to derivative martingales and decorated Poisson processes

Abstract

Multiplicative cascades, under weak or strong disorder, refer to sequences of positive random measures $\mu_{n,\beta}, n = 1,2,\dots$, parameterized by a positive disorder parameter $\beta$, and defined on the Borel $\sigma$-field ${\mathcal B}$ of $\partial T = \{0,1,\dots b-1\}^\infty$ for the product topology. The normalized cascade is defined by the corresponding sequence of random probability measures $prob_{n,\beta}:= Z_{n,\beta}^{-1}\mu_{n,\beta}, n = 1,2\dots,$ normalized to a probability by the partition function $Z_{n,\beta}$. In this note, a recent result of Madaule (2011) is used to explicitly construct a family of tree indexed probability measures $prob_{\infty,\beta}$ for strong disorder parameters $\beta > \beta_c$, almost surely defined on a common probability space. Moreover, viewing $\{prob_{n,\beta}: \beta > \beta_c\}_{n=1}^\infty$ as a sequence of probability measure valued stochastic process leads to finite dimensional weak convergence in distribution to a probability measure valued process $\{prob_{\infty,\beta}: \beta > \beta_c\}$. The limit process is constructed from the tree-indexed random field of derivative martingales, and the Brunet-Derrida-Madaule decorated Poisson process. A number of corollaries are provided to illustrate the utility of this construction.