# Optimal Uniform Convergence Rates and Asymptotic Normality for Series Estimators Under Weak Dependence and Weak Conditions ∗ 

First version January 2012; Revised August 2013, August 2014 

Xiaohong Chen † and Timothy M. Christensen ‡ 

## Abstract 

We show that spline and wavelet series regression estimators for weakly dependent regressors attain the optimal uniform (i.e. sup-norm) convergence rate ( n/ log n ) − p/ (2 p + d ) of Stone (1982), where d is the number of regressors and p is the smoothness of the regression function. The optimal rate is achieved even for heavy-tailed martingale difference errors with finite (2 + ( d/p ))th absolute moment for d/p < 2. We also establish the asymptotic normality of t statistics for possibly nonlinear, irregular functionals of the conditional mean function under weak conditions. The results are proved by deriving a new exponential inequality for sums of weakly dependent random matrices, which is of independent interest. 

JEL Classification: C12, C14, C32 

Keywords: Nonparametric series regression; Optimal uniform convergence rates; Weak depen- dence; Random matrices; Splines; Wavelets; (Nonlinear) Irregular Functionals; Sieve t statistics 

