Uniform convergence of sample second moments of families of time series arrays

Abstract

We consider abstractly defined time series arrays yt(T), 1 ≤ t ≤ T, requiring only that their sample lagged second moments converge and that their end values y1+j(T) and yT-j(T) be of order less than T 1/2 for each j ≥ 0. We show that, under quite general assumptions, various types of arrays that arise naturally in time series analysis have these properties, including regression residuals from a time series regression, seasonal adjustments and infinite variance processes rescaled by their sample standard deviation. We establish a useful uniform convergence result, namely that these properties are preserved in a uniform way when relatively compact sets of absolutely summable filters are applied to the arrays. This result serves as the foundation for the proof, in a companion paper by Findley, Pötscher and Wei, of the consistency of parameter estimates specified to minimize the sample mean squared multistep-ahead forecast error when invertible short-memory models are fit to (short-or long-memory) time series or time series arrays.