A cascade decomposition theory with applications to Markov and exchangeable cascades

Abstract

A multiplicative random cascade refers to a positive T T -martingale in the sense of Kahane on the ultrametric space T = { 0 , 1 , … , b − 1 } N . T = { \{ 0,1,\dots ,b-1 \} }^{\mathbf {N}}. A new approach to the study of multiplicative cascades is introduced. The methods apply broadly to the problems of: (i) non-degeneracy criterion, (ii) dimension spectra of carrying sets, and (iii) divergence of moments criterion. Specific applications are given to cascades generated by Markov and exchangeable processes, as well as to homogeneous independent cascades.