Uniform convergence of sample second moments of families of time series arrays. (English) Zbl 1041.62073

Findley, David F.; Pötscher, Benedikt M.; Wei, Ching-Zong

Ann. Stat. 29, No. 3, 815-838 (2001).

Summary: We consider abstractly defined time series arrays \(y_t(T),\;1\leq t\leq T\), requiring only that their sample lagged second moments converge and that their end values \(y_{1+j}(T)\) and \(y_{T-j}(T)\) be of order less than \(T^{1/2}\) for each \(j\geq0\). 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 our companion paper, J. Econom. 118, 151–187 (2004; Zbl 1033.62092), 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.

MSC:

62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62E20	Asymptotic distribution theory in statistics
60F15	Strong limit theorems

Keywords:

regression residuals; lacunary systems; infinite variance processes; long memory processes; seasonally adjusted series; locally stationary series; uniform laws of large numbers; consistency

Citations:

Zbl 1033.62092

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