## Abstract

Multiplicative cascades, under weak or strong disorder, refer to sequences of positive random measures µ_{n,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 prob_{n,β}:= Z^{-1}_{n,β}µ_{n,β}, n = 1, 2..., normalized to a probability by the partition function Z_{n,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 > β_{c}, almost surely defined on a common probability space. Moreover, viewing {prob_{n,β}: β > β_{c}}^{∞}_{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 > β_{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.

Original language | English (US) |
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Article number | 32 |

Journal | Electronic Communications in Probability |

Volume | 20 |

DOIs | |

State | Published - Apr 1 2015 |

## Keywords

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

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty