Assessing the number of mean square derivatives of a Gaussian process

We consider a real Gaussian process X with unknown smoothness where the mean square derivative X(r0) is supposed to be Hölder continuous in quadratic mean. First, from selected sampled observations, we study the reconstruction of X(t), t[set membership, variant][0,1], with 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 of r0 and prove its strong consistency by giving an exponential inequality for . Finally, we establish the strong consistency of with an almost optimal rate.