Asymptotics of multifractal products of spherical random fields

TL;DR

This work addresses the construction of multifractal random measures on the sphere $\mathbb{S}^d$ as limits of multifractal products of spherical fields. It develops limit theorems ensuring convergence of the cumulative measures $\mu_n$ under general mixing conditions and derives the Rényi function of the limiting measure. The results include non-degeneracy criteria and explicit Renyi-function expressions, with a Gaussian/log-Gaussian specialization yielding a quadratic form in $q$. The framework broadens applicability to diverse spherical phenomena (e.g., cosmology, geophysics) by enabling flexible multifractal properties through general mixing assumptions.

Abstract

The paper studies multifractal random measures on the sphere $\mathbb{S}^d$ constructed via multifractal products of random fields. It presents new limit theorems for multifractal products of spherical fields and conditions for the non-degeneracy of the limiting measure. The multifractal properties of the limiting measure are investigated, and its Rényi function is derived. Compared to earlier results on multifractal products of spherical fields, the obtained limit theorems hold under general mixing conditions, enabling the consideration of multifractal products of fields from a broad class and the construction of random measures with flexible multifractal properties.

Theorems & Definitions (17)

• Lemma 1
• Lemma 2
• proof
• Theorem 1
• proof
• Corollary 1
• Remark 1
• Theorem 2
• proof
• Corollary 2