A Cascade Decomposition Theory with Applications to Markov and Exchangeable Cascades

A multiplicative random cascade refers to a positive T -martingale in the sense of Kahane on the ultrametric space T = {0, 1, . . . , b− 1}. 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. 1. Positive T -martingales Positive T-martingales were introduced by Jean-Pierre Kahane as the general framework for independent multiplicative cascades and random coverings. Although originating in statistical theories of turbulence, the general framework also includes certain spin-glass and random polymer models as well as various other spatial distributions of interest in both probability theory and the physical sciences. For basic definitions, let T be a compact metric space with Borel sigmafield B, and let (Ω,F , P ) be a probability space together with an increasing sequence Fn, n = 1, 2, . . . , of sub-sigmafields of F . A positive T-martingale is a sequence {Qn} of B × F−measurable non-negative functions on T × Ω such that (i) For each t ∈ T, {Qn(t, ·) : n = 0, 1, . . . } is a martingale adapted to Fn, n = 0, 1, . . . ; (ii) For P -a.s. ω ∈ Ω, {Qn(·, ω) : n = 0, 1, . . . } is a sequence of Borel measurable non-negative real-valued functions on T . Let M(T ) denote the space of positive Borel measures on T and suppose that {Qn(t)} is a positive T -martingale. For σ ∈ M(T ) such that q(t) := EQn(t) ∈ L(σ), let σn ≡ Qnσ denote the random measure defined by Qnσ << σ and dQnσ dσ (t) := Qn(t), t ∈ T. Then, essentially by the martingale convergence theorem, one obtains a random Borel measure σ∞ ≡ Q∞σ such that for f ∈ C(T ), (1.1) lim n→∞ ∫ T f(t)Qn(t, ω)σ(dt) = ∫ T f(t)Q∞σ(dt, ω) a.s. Received by the editors August 18, 1994. 1991 Mathematics Subject Classification. Primary 60G57, 60G30, 60G42; Secondary 60K35, 60D05, 60J10, 60G09.