Optimal Uniform Convergence Rates and Asymptotic Normality for Series Estimators Under Weak Dependence and Weak Conditions ∗ Xiaohong Chen † and Timothy M. Christensen ‡ First version January 2012; Revised August 2013, August 2014 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. Keywords: Nonparametric series regression; Optimal uniform convergence rates; Weak depen- dence; Random matrices; Splines; Wavelets; (Nonlinear) Irregular Functionals; Sieve t statistics JEL Classification: C12, C14, C32 $^{∗}$We thank the guest coeditor, two anonymous referees, Bruce Hansen, Jianhua Huang, Stefan Schneeberger, and conference participants of SETA2013 in Seoul and AMES2013 in Singapore for useful comments. This paper is an expanded version of Sections 4 and 5 of our Cowles Foundation Discussion Paper No. 1923 (Chen and Christensen, 2013), the remaining parts of which are currently undergoing a major revision. Support from the Cowles Foundation is gratefully acknowledged. Any errors are the responsibility of the authors. $^{†}$Corresponding Author. Cowles Foundation for Research in Economics, Yale University, Box 208281, New Haven, CT 06520, USA. Tel: +1 203 432 5852; fax: +1 203 432 6167. E-mail address: xiaohong.chen@yale.edu $^{‡}$Department of Economics, New York University, 19 West 4th Street, New York, NY 10012, USA. E-mail address: timothy.christensen@nyu.edu 1 arXiv:1412.6020v2 [math.ST] 10 Mar 2015 Abstract