A Limit Theorem for Maxima of Functions of Gaussian Processes with Logarithmic Decay of Correlation

Limit theorems for the maxima of functions of Gaussian time series are studied. The limit behavior of the normalized sequence of maxima is examined under the condition that the correlation function of the process under consideration decays strictly logarithmically. Under some reasonable constraints on the function under consideration, the distribution is shown to be a modification of the corresponding distribution from Gnedenko's theorem. In addition, we derive a limit theorem for the reliability index of the vector function of a dependent vector of standard normal random variables, in which each component has the distribution function from the attraction domain of the Fréchet distribution. This result is obtained under the assumption that the correlation function of each component of the Gaussian vector decays at least logarithmically.