ESTIMATING THE SINGULARITY FUNCTION OF A GAUSSIAN PROCESS WITH APPLICATIONS

A non-parametric estimator of the singularity function of a discretely observed Gaussian process on [0, 1] is built, using projection kernel s on [0, 1], This estimator is based on a generalized quadratic variat ion procedure, A first asymptotic study is done with respect to the in tegrated mean square error, for which we find the classical non-parame tric rate of convergence. In a second asymptotic study, we prove weak convergence in distribution of our estimator, suitably normalized. The se results are applied to two related topics: estimation of the mean s quare error in estimating linear functionals of a random process; and estimation of the diffusion coefficient in a diffusion model.