ON PROBABILITY OF HIGH EXTREMES FOR PRODUCT OF TWO GAUSSIAN STATIONARY PROCESSES

Let (X(t), Y (t)), t >= 0, be a zero-mean stationary Gaussian vector process with a covariance functions for components r(i)(t) satisfying Pickand's condition r(i)(t) = 1-c(i)vertical bar t vertical bar(alpha i) (1+ o(1)), t -> 0, c(i) > 0, 0 < alpha(i) <= 2, i = 1, 2. Let r(i)(t) < 1, i = 1, 2, t > 0. Assuming that r equivalent to EX(t) Y (t) is an element of (-1, 1) and lim(t,s -> 0)(EX(t) Y (s) - r)/vertical bar t - s vertical bar min(alpha(1), alpha(2)) exists, we study the behavior of probability P(max(t is an element of)[0, p] X(t) Y (t) > u) as u ->infinity for any p. In particular, we derive for any p the exact asymptotic behavior of probability P(max(t is an element of)[0, p](X-2(t) -Y-2(t)) > u) as u -> infinity for independent Gaussian stationary processes X(t), Y (t) satisfying the above conditions.