EXTREMES AND CROSSINGS FOR DIFFERENTIABLE STATIONARY-PROCESSES WITH APPLICATION TO GAUSSIAN-PROCESSES IN RM AND HILBERT-SPACE

Let {omega(t)}t greater-than-or-equal-to 0 be a stochastically differentiable stationary process in R(m) and let A(u) subset-or-equal-to R(m) satisfy lim(u up u2) P{omega(0) is-an-element-of A(u)} = 0. We give a method to find the asymptotic behaviour of P{or 0 less-than-or-equal-to t less-than-or-equal-to h {omega(t) is-an-element-of A(u)}} as u up u2. We use our method to study hitting probabilities for small sets with application to Gaussian processes and to study suprema of processes in R with application to (the norm of) Gaussian processes in Hilbert space.