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Gaussian processes are widely employed as versatile modelling and predictivetools in spatial statistics, functional data analysis, computer modelling anddiverse applications of machine learning. They have been widely studied overEuclidean spaces, where they are specified using covariance functions orcovariograms for modelling complex dependencies. There is a growing literatureon Gaussian processes over Riemannian manifolds in order to develop richer andmore flexible inferential frameworks for non-Euclidean data. While numericalapproximations through graph representations have been well studied for theMat\'ern covariogram and heat kernel, the behaviour of asymptotic inference onthe parameters of the covariogram has received relatively scant attention. Wefocus on the asymptotic inference for Gaussian processes constructed overcompact Riemannian manifolds. Building upon the recently introduced Mat\'erncovariogram on a compact Riemannian manifold, we employ formal notions andconditions for the equivalence of two Mat\'ern Gaussian random measures oncompact manifolds to derive the parameter that is identifiable, also known asthe microergodic parameter, and formally establish the consistency of themaximum likelihood estimate and the asymptotic optimality of the best linearunbiased predictor. The circle is studied as a specific example of compactRiemannian manifolds with numerical experiments to illustrate and corroboratethe theory.

Inference for Gaussian Processes with Mat\'ern Covariogram on Compact Riemannian Manifolds