Bounds on the support of the multifractal spectrum of stochastic processes

TL;DR

This paper establishes bounds on the support of the multifractal spectrum of stochastic processes, revealing the influence of moments and tail behavior on the spectrum's properties.

Contribution

It introduces bounds on the spectrum support, complements Kolmogorov's criterion, and analyzes the role of moments and tail behavior in determining the spectrum.

Findings

Negative moments do not affect the spectrum in ergodic self-similar processes.

Infinite positive moments lead to a nontrivial spectrum.

Heavy-tailed processes are necessary for a nontrivial spectrum.

Abstract

Multifractal analysis of stochastic processes deals with the fine scale properties of the sample paths and seeks for some global scaling property that would enable extracting the so-called spectrum of singularities. In this paper we establish bounds on the support of the spectrum of singularities. To do this, we prove a theorem that complements the famous Kolmogorov's continuity criterion. The nature of these bounds helps us identify the quantities truly responsible for the support of the spectrum. We then make several conclusions from this. First, specifying global scaling in terms of moments is incomplete due to possible infinite moments, both of positive and negative order. For the case of ergodic self-similar processes we show that negative order moments and their divergence do not affect the spectrum. On the other hand, infinite positive order moments make the spectrum nontrivial. In particular, we show that the self-similar stationary increments process with the nontrivial spectrum must be heavy-tailed. This shows that for determining the spectrum it is crucial to capture the divergence of moments. We show that the partition function is capable of doing this and also propose a robust variant of this method for negative order moments.