Assessing the number of mean square derivatives of a Gaussian process

We consider a real Gaussian process X with unknown smoothness r 0 ∈ N 0 where the mean square derivative X ( r 0 ) is supposed to be Hölder continuous in quadratic mean. First, from selected sampled observations, we study the reconstruction of X ( t ) , t ∈ [ 0 , 1 ] , with X ˜ r ( 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 ≥ r 0 . Next, from an interpolation-based empirical criterion and n sampled observations of X , we derive an estimator r ̂ n of r 0 and prove its strong consistency by giving an exponential inequality for P ( r ̂ n ≠ r 0 ) . Finally, we establish the strong consistency of X ˜ max ( r ̂ n , 1 ) ( t ) with an almost optimal rate.