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

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