  Gaussian processes are widely employed as versatile modelling and predictive
tools in spatial statistics, functional data analysis, computer modelling and
diverse applications of machine learning. They have been widely studied over
Euclidean spaces, where they are specified using covariance functions or
covariograms for modelling complex dependencies. There is a growing literature
on Gaussian processes over Riemannian manifolds in order to develop richer and
more flexible inferential frameworks for non-Euclidean data. While numerical
approximations through graph representations have been well studied for the
Mat\'ern covariogram and heat kernel, the behaviour of asymptotic inference on
the parameters of the covariogram has received relatively scant attention. We
focus on the asymptotic inference for Gaussian processes constructed over
compact Riemannian manifolds. Building upon the recently introduced Mat\'ern
covariogram on a compact Riemannian manifold, we employ formal notions and
conditions for the equivalence of two Mat\'ern Gaussian random measures on
compact manifolds to derive the parameter that is identifiable, also known as
the microergodic parameter, and formally establish the consistency of the
maximum likelihood estimate and the asymptotic optimality of the best linear
unbiased predictor. The circle is studied as a specific example of compact
Riemannian manifolds with numerical experiments to illustrate and corroborate
the theory.
