We consider covariance parameter estimation for Gaussian processes with functional inputs. From an increasing-domain asymptotics perspective, we prove the asymptotic consistency and normality of the maximum likelihood estimator. We extend these theoretical guarantees to encompass scenarios accounting for approximation errors in the inputs, which allows robustness of practical implementations relying on conventional sampling methods or projections onto a functional basis. Loosely speaking, both consistency and normality hold when the approximation error becomes negligible, a condition that is often achieved as the number of samples or basis functions becomes large. These later asymptotic properties are illustrated through analytical examples, including one that covers the case of non-randomly perturbed grids, as well as several numerical illustrations.

翻译：我们考虑函数型输入高斯过程的协方差参数估计问题。从递增域渐近视角出发，我们证明了极大似然估计的渐近一致性和正态性。我们将这些理论保障扩展到涵盖输入近似误差的情形，这使得依赖传统采样方法或函数基投影的实际实现具有鲁棒性。大致而言，当近似误差可忽略时（该条件通常随样本量或基函数数量增大而实现），一致性和正态性均成立。我们通过解析示例（包括非随机扰动网格情形）及多项数值实验阐明了这些渐近性质。