Assessing the number of mean-square derivatives of a Gaussian process

We consider a real Gaussian process X with unknown smoothness r0 ∈ N0 where the mean-square derivative X(r0) is supposed to be Hölder continuous in quadratic mean. First from selected sampled observations, we study reconstruction of X(t), t ∈ [0, 1] with X̃r(t), a piecewise polynomial interpolation of degree r ≥ 1. We show that the mean-square error of the interpolation is a decreasing function of r but becomes stable as soon as r ≥ r0. Next, from an interpolation-based empirical criterion and n sampled observations of X, we derive an estimator r̂n of r0 and prove its strong consistency by giving an exponential inequality for P(r̂n 6= r0). Finally, we establish the strong consistency of X̃max(r̂n,1)(t) with an almost optimal rate.