Random wavelet series based on a tree-indexed Markov chain

TL;DR

This paper analyzes the multifractal properties of random wavelet series with correlated coefficients modeled by a tree-indexed Markov chain, revealing their spectrum of singularities and oscillating behaviors.

Contribution

It introduces a novel multifractal analysis of wavelet series with Markov chain correlations, characterizing their singularity spectrum and local oscillations.

Findings

Determines the spectrum of singularities for these series.

Shows almost every sample path has oscillating singularities.

Identifies sets with large intersection for points with bounded Holder exponents.

Abstract

We study the global and local regularity properties of random wavelet series whose coefficients exhibit correlations given by a tree-indexed Markov chain. We determine the law of the spectrum of singularities of these series, thereby performing their multifractal analysis. We also show that almost every sample path displays an oscillating singularity at almost every point and that the points at which a sample path has at most a given Holder exponent form a set with large intersection.