Multifractal analysis in a mixed asymptotic framework

TL;DR

This paper extends multifractal analysis of random cascades to a mixed asymptotic framework, allowing for a continuous interpolation between single sample and infinite realizations, with new convergence and limit theorems.

Contribution

It introduces a mixed asymptotic approach for multifractal analysis, establishing convergence properties, a central limit theorem, and a multifractal formalism within this new framework.

Findings

Scaling exponents depend on the mixed asymptotic exponent .

Established a central limit theorem for partition functions.

Validated results within a wavelet analysis framework.

Abstract

Multifractal analysis of multiplicative random cascades is revisited within the framework of {\em mixed asymptotics}. In this new framework, statistics are estimated over a sample which size increases as the resolution scale (or the sampling period) becomes finer. This allows one to continuously interpolate between the situation where one studies a single cascade sample at arbitrary fine scales and where at fixed scale, the sample length (number of cascades realizations) becomes infinite. We show that scaling exponents of ''mixed'' partitions functions i.e., the estimator of the cumulant generating function of the cascade generator distribution, depends on some ``mixed asymptotic'' exponent $\chi$ respectively above and beyond two critical value $p_\chi^-$ and $p_\chi^+$. We study the convergence properties of partition functions in mixed asymtotics regime and establish a central limit theorem. These results are shown to remain valid within a general wavelet analysis framework. Their interpretation in terms of Besov frontier are discussed. Moreover, within the mixed asymptotic framework, we establish a ``box-counting'' multifractal formalism that can be seen as a rigorous formulation of Mandelbrot's negative dimension theory. Numerical illustrations of our purpose on specific examples are also provided.