On normalized multiplicative cascades under strong disorder

Partha S. Dey, Edward C. Waymire

Research output: Contribution to journalArticlepeer-review

Abstract

Multiplicative cascades, under weak or strong disorder, refer to sequences of positive random measures µn,β, n = 1, 2,..., parameterized by a positive disorder parameter β, and defined on the Borel σ-field ß of ∂T = {0, 1,... b - 1} for the product topology. The normalized cascade is defined by the corresponding sequence of random probability measures probn,β:= Z-1n,βµn,β, n = 1, 2..., normalized to a probability by the partition function Zn,β. In this note, a recent result of Madaule [27, 2011] is used to explicitly construct a family of tree indexed probability measures prob∞,β for strong disorder parameters β > βc, almost surely defined on a common probability space. Moreover, viewing {probn,β: β > βc}n=1 as a sequence of probability measure valued stochastic process leads to finite dimensional weak convergence in distribution to a probability measure valued process {prob∞,β : β > βc}. 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.

Original languageEnglish (US)
Article number32
JournalElectronic Communications in Probability
Volume20
DOIs
StatePublished - Apr 1 2015

Keywords

  • Multiplicative cascade
  • Partition function
  • Strong disorder
  • Tree polymer

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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