REGULARIZED GAUSSIAN-PROCESSES, CROSSINGS AND LOCAL TIME

Let { X(t), 1 is-an-element-of [0, 1] } be a stationary, centered Gaussian process defined on (OMEGA, A, P), with covariance function r satisfying: r (t) approximately 1 - C Absolute value of t 2alpha, 0 < t < delta, 0 < alpha < 1. We define the regularized process X(epsilon) = psi(epsilon)*X and Y(epsilon) = X(epsilon)/sigma(epsilon) where sigma(epsilon)2 = var X(t)epsilon, with psi(epsilon) a kernel that approaches Dirac's delta function. We study the convergence of Z(epsilon) (f) = epsilon(-a (alpha)) integral-+infinity/-infinity [(N(Yepsilon) (x)/c (epsilon)) - L(X) (x)] f (x) dx when epsilon goes to zero, N(Yepsilon) (x) is the number of crossings for Y(epsilon) at level x in [0, 1] and L(X) (x) is the local time of X in x on [0, 1]. The limit depends on the value of alpha.