APPROXIMATION OF GAUSSIAN LOCAL TIME

Let X be a centered stationary Gaussian stochastic process with a d-di mensional parameter (d greater-than-or-equal-to 2), F its spectral mea sure, integral R(d) parallel-to x parallel-to 2 F(dx) = +infinity (par allel-to x parallel-to denotes the Euclidean norm of x). We consider r egularizations of the trajectories of X by means of convolutions of th e form X(epsilon)(t)=(PSI(epsilon)X)(t) where PSI(epsilon) stands for an approximation of unity (as epsilon tends to zero) satisfying certa in regularity conditions. The aim of this paper is to recover the loca l time of X at a given level u, as a limit of appropriate normalizatio ns of the geometric measure of the u-level set of the regular approxim ating processes X(epsilon). A part of the difficulties comes from the fact that the geometric behavior of the covariance of the Gaussian pro cess X(epsilon) can be a complex one as epsilon approaches 0. The resu lts are on L2-convergence and include bounds for the speed of converge nce. L(p) results may be obtained in similar ways, but almost sure con vergence or simultaneous convergence for the various values of u do no t seem to follow from our methods. In Sect. 3 we have included example s showing a diversity of geometric behaviors, especially in what conce rns the dependence on the thickness of the set in which the covariance of the original process X is irregular.Some technical results of anal ytic nature are included as appendices in Sect. 4.