Mean square consistency and improved rate of convergence of generalized subsampling estimator for non-stationary time series

In this article, we show the mean square consistency for a generalized subsampling estimator based on the aggregation of the mean, median, and trimmed mean of some subsampling estimators for general non-stationary time series. Consistency requires standard assumptions, including the existence of moments and alpha-mixing conditions. We apply our results to the Fourier coefficients of the autocovariance function of periodically correlated time series. Furthermore, as in the i.i.d. case, we show that the generalized subsampling estimator satisfies Bernstein inequality and concentrates at an improved rate (under the condition of no or small bias) compared with the original estimator. Finally, we illustrate our results with some simulation data examples.