Equivalent Gaussian distributions on commutative spaces: An RKHS analysis

Abstract

The investigation of equivalent Gaussian distributions for stochastic processes is a central problem in probability and statistics. In this context, the choice of the index set and the correlation structure, particularly their interaction, plays a crucial role. The purpose of this paper is to show how an explicit description of the corresponding reproducing kernel Hilbert space (RKHS) helps to better understand this interplay. In the stationary setting, when the index set is taken to be a homogeneous space, we show how an RKHS approach allows us to bridge the gap to harmonic analysis on commutative spaces, thereby further complementing the characterization of equivalent Gaussian distributions via their spectral measures.