  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.
