Weak convergence of a nonlinear functional of a stationary Gaussian process.: Application to the local time

Let {X-t 0 less than or equal to t less than or equal to 1} be a centered stationary Gaussian process, with correlation function satisfying the condition p(t) = 1 - t(beta)L(t), 0 < beta < 2, and let L be a slowly varying function at zero. Observing the process at points i/N, i = 0, i,..., N and considering |X-i/N - X(i-1)/N|(P) with p > 0, We study the properties of the Donsker line associated with p-th order variations (i=1)Sigma([Nt]) |X-i/N - X(i-1)/N|(P) We also study the relationship between the number of crossings of a regularization of the initial process and the local time of the initial process. The results depend on the values of beta.