Does randomness in multifractals imply latent dimensions?

TL;DR

This paper investigates how randomness affects the presence of latent dimensions in multifractals, revealing conditions under which these dimensions may be absent in discrete random multinomial measures.

Contribution

It introduces the latent dimensions condition (LDC) and analyzes the asymptotic behavior of the multifractal function in random multinomial measures.

Findings

Latent dimensions can be absent under certain conditions in discrete random multinomial measures.

The asymptotic behavior of the multifractal function is characterized.

Examples illustrate the impact of randomness on latent dimensions.

Abstract

Negative, or latent, dimensions have always attracted a strong interest since their discovery. When randomness is introduced in multifractals, the sample-to-sample fluctuations of multifractal spectra emerge inevitably, which has motivated various studies in this field. In this work, we study a class of multinomial measures and argue the asymptotic behaviors of the multifractal function as . The so-called latent dimensions condition (LDC) is presented which states that latent dimensions may be absent in discrete random multinomial measures. In order to clarify the discovery, several examples are illustrated.