Approximation du temps local des processus gaussiens stationnaires par régularisation des trajectoires. (Approximation of local times of stationary Gaussian processes by regularization of trajectories) (Q1085881)

Let \(\{\) X(t)\(\}\) be a stationary non-differentiable Gaussian process and let \(\phi _{\epsilon}(u)=\epsilon ^{-1}\phi (u/\epsilon)\) be an approximate identity. Setting \(X_{\epsilon}(t)=X^ *\phi _{\epsilon}(t)\) and letting \(N_{\epsilon}(T)\) be the number of zeros of \(X_{\epsilon}\) in the interval [0,T] it is shown that under weak technical conditions there are constants C(\(\epsilon)\) so that \(C(\epsilon)N_{\epsilon}(T)\) converges in \(L^ 2\) as \(\epsilon\) \(\to 0\). When X admits a continuous local time, the limit is the local time L(0,T) at zero of X(t).