Extremes of Gaussian processes with a smooth random variance

Let ξ(t)ξ(t) be a standard stationary Gaussian process with covariance function r(t)r(t), and η(t)η(t), another smooth random process. We consider the probabilities of exceedances of ξ(t)η(t)ξ(t)η(t) above a high level uu occurring in an interval [0,T][0,T] with T>0T>0. We present asymptotically exact results for the probability of such events under certain smoothness conditions of this process ξ(t)η(t)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)X(t) given a random element YY.

► Stationary Gaussian processes multiplied with another random process are investigated.
► We consider exceedances of this process above a level u in the time interval [0,T][0,T].
► Exact asymptotical results for the probability of such events are derived for large uu.
► A general class of conditional Gaussian processes given a random element YY is defined.
► A large deviation result is shown for this general class.