Open Access
June 2001 Uniform convergence of sample second moments of families of time series arrays
David F. Findley, Benedikt M. Pötscher, Ching-Zong Wei
Ann. Statist. 29(3): 815-838 (June 2001). DOI: 10.1214/aos/1009210691


We consider abstractly defined time series arrays y t(T), 1 \le t\le 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 T½ for each j \ge 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.


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David F. Findley. Benedikt M. Pötscher. Ching-Zong Wei. "Uniform convergence of sample second moments of families of time series arrays." Ann. Statist. 29 (3) 815 - 838, June 2001.


Published: June 2001
First available in Project Euclid: 24 December 2001

zbMATH: 1041.62073
MathSciNet: MR1865342
Digital Object Identifier: 10.1214/aos/1009210691

Primary: 62M10 , 62M15 , 62M20
Secondary: 60G10 , 62J05

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

Rights: Copyright © 2001 Institute of Mathematical Statistics

Vol.29 • No. 3 • June 2001
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