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

We consider abstractly defined time series arrays y(t)(T), 1 less than or e qual to t less than or equal to T, requiring only that their sample lagged second moments converge and that their end values y(1+j)(T) and y(T-j)(T) b e of order less than T-1/2 for each j greater than or equal to 0. We show t hat, under quite general assumptions, various types of arrays that arise na turally 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 establi sh a useful uniform convergence result, namely that these properties are pr eserved in a uniform way when relatively compact sets of absolutely summabl e filters are applied to the arrays. This result serves as the foundation f or the proof, in a companion paper by Findley, Potscher and Wei, of the con sistency of parameter estimates specified to minimize the sample mean squar ed multistep-ahead forecast error when invertible short-memory models are f it to (short- or long-memory) time series or time series arrays.