Noncentral limit results for spatiotemporal random fields on manifolds and beyond

This paper derives noncentral limit results (NCLTs) for suitable scaling of functionals of spatially homogeneous and isotropic, and stationary in time, LRD Gaussian subordinated Spatiotemporal Random Fields (STRFs) with Hermite rank equal to two. The cases of connected and compact two point homogeneous spaces M_{d} in R^{d+1}, and compact convex sets K in R^{d+1},$ whose interior has positive Lebesgue measure, are analyzed. These NCLTs are obtained in the second Wiener Chaos by applying reduction theorems. The methodological approaches adopted in the derivation of these results are based on the pure point and continuous spectra of the Gaussian STRFs subordinators defined on M_{d} and K, respectively.

💡 Research Summary

The manuscript establishes non‑central limit theorems (NCLTs) for functionals of long‑range dependent (LRD) Gaussian‑subordinated spatiotemporal random fields (STRFs) that are homogeneous and isotropic in space and stationary in time, focusing on the case where the Hermite rank of the nonlinear transformation equals two. Two geometric settings are considered: (i) connected, compact two‑point homogeneous manifolds (M_d) embedded in (\mathbb{R}^{d+1}) (with the sphere as the canonical example) and (ii) compact convex subsets (K\subset\mathbb{R}^{d+1}) whose interior has positive Lebesgue measure.

The authors first introduce a Gaussian STRF (Z(x,t)) with covariance kernel (C_Z). By restricting (Z) to an expanding spatiotemporal domain (M_d(T^\gamma)\times T(T)) (where (T^\gamma) denotes a homothetic spatial scaling and (T(T)=