Convergence of complex multiplicative cascades

TL;DR

This paper extends the theory of cascade measures to include complex weights, analyzing their convergence and limit behaviors, and establishing conditions for multifractal modeling and connections to Brownian motion.

Contribution

It introduces complex-valued weights into cascade measures, providing new convergence criteria and characterizing their multifractal limits and stochastic behaviors.

Findings

Established a sufficient condition for almost sure uniform convergence to self-similar limits.

Identified conditions under which the limit is monofractal or trivial.

Proved a functional central limit theorem leading to Brownian motion in multifractal time.

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.