Asymptotics of distributions of maxima of random processes

We consider problems of finding the asymptotics of distribution of maximum of nonstationary Gaussian and non-Gaussian processes. In the paper by V. I. Piterbarg in 1988 there was proved a theorem on the maximum of a centered locally stationary Gaussian process with variance attaining maximum at a unique point. It states that the asymptotics of the distribution is determined by the ratio of the powers of the expansions of the correlation function and variance near the maximum point. A natural question arises: is the power-law behavior of the expansions of these functions essential, or is it sufficient to be able to compare the convergence rates of the variance and correlation at a given point? In the first part we answer this question. An important question is the shape of the trajectories of Gaussian exceedances of a high level, which is discussed in the second part. The third part is devoted to the Cramer-Leadbetter conditions for strong mixing and how it can be effectively applied to copula processes. In conclusion we consider a discussion of the Kolmogorov problem solved by Prokhorov and how it can be generalized to Gaussian and copula Gaussian sequences.