Assessing the number of mean square derivatives of a Gaussian process

We consider a real Gaussian process XX with unknown smoothness r0∈N0r0∈N0 where the mean square derivative X(r0)X(r0) is supposed to be Hölder continuous in quadratic mean. First, from selected sampled observations, we study the reconstruction of X(t)X(t), t∈[0,1]t∈[0,1], with X˜r(t) a piecewise polynomial interpolation of degree r≥1r≥1. We show that the mean square error of the interpolation is a decreasing function of rr but becomes stable as soon as r≥r0r≥r0. Next, from an interpolation-based empirical criterion and nn sampled observations of XX, we derive an estimator r̂n of r0r0 and prove its strong consistency by giving an exponential inequality for P(r̂n≠r0). Finally, we establish the strong consistency of X˜max(r̂n,1)(t) with an almost optimal rate.