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.

翻译：高斯过程被广泛用作空间统计、函数型数据分析、计算机建模及机器学习多种应用中的通用建模与预测工具。这些方法在欧氏空间中得到充分研究，可通过协方差函数或协变异函数刻画复杂依赖关系。为发展更丰富灵活的非欧氏数据推断框架，关于黎曼流形上高斯过程的文献正日益增多。尽管通过图表示进行数值逼近的方法在Matérn协变异函数和热核中已得到充分研究，但协变异函数参数的渐近推断行为却鲜受关注。本文聚焦于紧致黎曼流形上构建的高斯过程的渐近推断。基于近期提出的紧致黎曼流形上Matérn协变异函数，我们利用两个Matérn高斯随机测度在紧致流形上等价的严格定义与条件，推导出可识别参数（亦称微遍历参数），并正式建立了极大似然估计的一致性以及最优线性无偏预测的渐近最优性。以圆环作为紧致黎曼流形的特例进行数值实验，以阐释并验证该理论。