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Laws of Large Numbers for Hilbert Space-Valued Mixingales with Applications

Published online by Cambridge University Press:  11 February 2009

Xiaohong Chen  and
Halbert White
Xiaohong Chen
Affiliation:
University of Chicago
Halbert White
Affiliation:
University of California, San Diego
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Abstract

To obtain consistency results for nonparametric estimators based on stochastic processes relevant in econometrics, we introduce the notions of Hilbert space-valued Lp mixingales and near-epoch dependent arrays, and we prove weak and strong laws of large numbers by using a new exponential inequality for Hilbert (H) space-valued martingale difference arrays. We follow Andrews (1988, Econometric Theory 4, 458–467), Hansen (1991, Econometric Theory 7, 213–221; 1992, Econometric Theory 8, 421–422), Davidson (1993, Statistics and Probability Letters 16,301–304), and de Jong (1995, Econometric Theory 11, 347–358), extending results for H = R and improving memory conditions in certain instances. We give as examples consistency results for series and kernel estimators.

Type
Research Article
Information
Econometric Theory , Volume 12 , Issue 2 , June 1996 , pp. 284 - 304
Copyright
Copyright © Cambridge University Press 1996

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