A limit theorem for Gaussian copulas with weak dependence

This article is devoted to the generalization of Gnedenko–Fisher–Tippet's theorem to the case of Gaussian copular time series with dependence. A multivariate generalization of such a problem in terms of reliability index is considered as well. It's shown that if Berman's condition on the correlation function and some restriction to the copula of interest are met, then the limit theorem coincides with the one in the independent case. Moreover, the normalizing coefficients remain the same. Some sufficient conditions on the correlation function of the pair of Gaussian time series have been obtained. These conditions allow to transfer the limit theorem from independent case for reliability index of Weibull-like copulas to the case with dependence.