Limit theorems for multifractal products of random fields

TL;DR

This paper establishes new limit theorems for the convergence of multifractal products of random fields, providing less restrictive conditions and applying to multidimensional cases, with implications for understanding multifractal measures.

Contribution

It introduces generalized limit theorems with simple conditions for multifractal random fields, including multidimensional cases and a new class of geometric -sub-Gaussian fields.

Findings

Established convergence conditions in L_q spaces.

Derived rate of convergence for cumulative fields.

Presented a new class of geometric -sub-Gaussian random fields.

Abstract

This paper investigates asymptotic properties of multifractal products of random fields. The obtained limit theorems provide sufficient conditions for the convergence of cumulative fields in the spaces $L_q.$ New results on the rate of convergence of cumulative fields are presented. Simple unified conditions for the limit theorems and the calculation of the R\'enyi function are given. They are less restrictive than those in the known one-dimensional results. The developed methodology is also applied to multidimensional multifractal measures. Finally, a new class of examples of geometric $\varphi$-sub-Gaussian random fields is presented. In this case, the general assumptions have a simple form and can be expressed in terms of covariance functions only.