Asymptotic behavior of point processes of exits of a Gaussian stationary sequence beyond high levels

We study the asymptotic behavior of point processes of exits of a Gaussian stationary sequence beyond a level tending to infinity more slowly than in the Poisson limit theorem for the number of exits. Convergence in variation of such point processes to a marked Poisson process is proved. The results of Yu. V. Prokhorov on the best approximation of the Bernoulli distribution by a mixture of Gaussian and Poisson distributions are applied. A. N. Kolmogorov proposed this problem in the early 1950s.