Modeling and fitting of time series with heavy distribution tails and strong time dependence by Gaussian time series

In the model of Gaussian copula time series with the tails of one-dimensional distributions belonging to the Fréchet maximum domain of attraction and the description of dependency based on Gaussian variables (see [A. E. Mazur and V. I. Piterbarg, Moscow Univ. Math. Bull., 70 (2015), pp. 197–201]), an estimator for the copula (which is a nonlinear function that takes Gaussian variables to variables from the Fréchet maximum domain of attraction) is built. This opens the way for statistical analysis of data time series with potentially heavy tails using the machinery of asymptotic analysis of Gaussian sequences. The consistency and asymptotic normality for this estimator are proved.