Convergence of complex multiplicative cascades

Abstract

The familiar cascade measures are sequences of random positive measures obtained on $[0,1]$ via $b$-adic independent cascades. To generalize them, this paper allows the random weights invoked in the cascades to take real or complex values. This yields sequences of random functions whose possible strong or weak limits are natural candidates for modeling multifractal phenomena. Their asymptotic behavior is investigated, yielding a sufficient condition for almost sure uniform convergence to nontrivial statistically self-similar limits. Is the limit function a monofractal function in multifractal time? General sufficient conditions are given under which such is the case, as well as examples for which no natural time change can be used. In most cases when the sufficient condition for convergence does not hold, we show that either the limit is 0 or the sequence diverges almost surely. In the later case, a functional central limit theorem holds, under some conditions. It provides a natural normalization making the sequence converge in law to a standard Brownian motion in multifractal time.Comment: Published in at http://dx.doi.org/10.1214/09-AAP665 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Multifractal Analysis of Complex Random Cascades

“…This paper deals with the multifractal formalism for functions and the multifractal analysis of a new class of statistically self-similar functions introduced in [7]. This class is the natural extension to continuous functions of the random measures introduced in [39] and considered as a fundamental example of multifractal signals model since the notion of multifractality has been explicitely formulated [23,21,22] (see also [35,24,16,45,5] for the multifractal analysis and thermodynamical interpretation of these measures).…”

“…It is also possible to use the alternative approach consisting in showing that F W can be represented as a monofractal functions in multifractal time [43,53], and then consider the exponent h (1) F rather than h F . It turns out that such a time change also exists in the random case under restrictive assumptions on W , which include the deterministic case (see [7]). This is useful because, as we said, our calculations showed that in general in the random case it seems difficult to exploit the wavelet transform of F to compute its singularity spectrum.…”

Moments for multidimensional Mandelbrot cascades

“…Comparing this result with Theorem 2.1, we discover that ρ n (α) ≤ [κ(α)EN] n ≤ p α−1 ρ n (α) for all n; therefore, p α−1 ρ n (α) < 1 implies that κ(α)EN < 1. 6 C. HUANG Now we consider the existence of the harmonic moments of Y , i.e. E(Y i ) −λ < ∞ for each i ∈ {1, 2, .…”

Multifractal scenarios for products of geometric Lévy-based stationary models

“…Sufficient conditions for non-degeneracy and degeneracy in a general situation and relevant examples are provided in [34] (Equations (18) and (19), respectively.) The condition for complete degeneracy is detailed in Theorem 3 of [34].…”

“…(see, e.g., [35,36,33,34,48,26,42,24,52,30,50]). There are many ways to construct random multifractal models ranging from simple binomial cascades to measures generated by branching processes and the compound Poisson process [33,34,24,53,30,17,18,9,50,46,56,53,52,19,40,32,54]. In [31] it is shown that Lévy processes (except Brownian motion and Poisson processes) are multifractal; but since the increments of a Lévy process are independent, this class excludes the effects of dependence structures.…”