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：我们考虑函数输入高斯过程的协方差参数估计。从递增域渐近分析的角度，我们证明了极大似然估计量的渐近一致性和正态性。我们将这些理论保证扩展到涵盖输入近似误差的场景，从而增强了依赖于常规采样方法或投影到函数基上的实际实现的鲁棒性。粗略地说，当近似误差变得可忽略时，一致性和正态性均成立，这一条件通常随着样本或基函数数量的增加而实现。这些后验渐近性质通过分析示例（包括涵盖非随机扰动网格的情况）以及若干数值示例加以说明。