Laws of Large Numbers for Hilbert Space-Valued Mixingales with Applications
Xiaohong Chen () and
Econometric Theory, 1996, vol. 12, issue 2, 284-304
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, 458â€“467), Hansen (1991, Econometric Theory 7, 213â€“221; 1992, Econometric Theory 8, 421â€“422), Davidson (1993, Statistics and Probability Letters 16,301â€“304), and de Jong (1995, Econometric Theory 11, 347â€“358), extending results for H = R and improving memory conditions in certain instances. We give as examples consistency results for series and kernel estimators.
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