Asymptotics of multifractal products of spherical random fields

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.

💡 Research Summary

The paper develops a rigorous framework for constructing multifractal random measures on the unit sphere (S^{d}) by iteratively multiplying independent non‑negative random fields. Starting from a strongly (or weakly) isotropic spherical random field (T(\theta)) with zero mean and a covariance function that depends only on the geodesic distance, the authors introduce a sequence of i.i.d. fields ({\Lambda^{(i)}(\theta)}_{i\ge0}) satisfying (E