Extremes of Gaussian processes with a smooth random variance

Let ξ ( t ) be a standard stationary Gaussian process with covariance function r ( t ) , and η ( t ) , another smooth random process. We consider the probabilities of exceedances of ξ ( t ) η ( t ) above a high level u occurring in an interval [ 0 , T ] with T > 0 . We present asymptotically exact results for the probability of such events under certain smoothness conditions of this process ξ ( t ) η ( t ) , which is called the random variance process. We derive also a large deviation result for a general class of conditional Gaussian processes X ( t ) given a random element Y .