Probabilities of High Extremes for a Gaussian Stationary Process in a Random Environment / A. O. Kleban and M. V. Korulin. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2017. № 1. P. 11-16 [Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 10-14].

Let \xi \left(t \right) be a zero-mean stationary Gaussian process with the covariance function r\left(t \right) of Pickands type, i.e., r(t)=1-|t|^{\alpha }+o(|t|^{\alpha }),\;t\to 0,\;0<\alpha \leq 2, and \eta \left(t \right), \zeta \left(t \right) be periodic random processes. The exact asymptotic behavior of the probabilities P(\max_{t\in[0,T]} \eta \left(t \right) \xi \left(t \right) > u), P(\max_{t\in[0,T]} \left(\xi \left(t \right) + \eta \left(t \right)\right) > u) and P(\max_{t\in[0,T]} \left(\eta \left(t \right) \xi \left(t \right) + \zeta \left(t \right)\right) > u) is obtained for u \to \infty for any T>0 and independent \xi \left(t \right), \eta \left(t \right), \zeta \left(t \right).

Key words: Gaussian process, random environment, high extremes probabilities, double sum method, Laplace asymptotic method.

№ 1/2017