and wavelet series LS estimators attain the optimal sup-norm rates, allowing for weakly dependent data and heavy tailed error terms. It also presents general sharp L $^{2}$-norm convergence rates of series LS estimators with an arbitrary basis under very mild conditions. Section 3 provides the asymptotic normality of sieve t statistics for possibly nonlinear functionals of h$_{0}$ . Section 4 provides new exponen- tial inequalities for sums of weakly dependent random matrices, and a reinterpretation of equivalence of the theoretical and empirical L 2 norms as a criterion regarding convergence of a certain random matrix. Section 5 shows the sup-norm stability of the empirical L 2 projections onto compactly sup- ported wavelet bases, which provides a tight upper bound on the sup-norm bias term for the wavelet series LS estimator. The results in Sections 4 and 5 are of independent interest. Section 6 contains a brief review of spline and wavelet sieve bases. Proofs and ancillary results are presented in Section 7. 

Notation: Let λ$_{min}$ ( · ) and λ$_{max}$ ( · ) denote the smallest and largest eigenvalues, respectively, of a matrix. The exponent − denotes the Moore-Penrose generalized inverse. ‖ · ‖ denotes the Euclidean norm when applied to vectors and the matrix spectral norm (i.e., largest singular value) when applied to matrices, and ‖ · ‖ ℓ p denotes the ℓ p norm when applied to vectors and its induced operator norm when applied to matrices (thus ‖ · ‖ = ‖ · ‖ ℓ 2 ). If { a$_{n}$ : n ≥ 1 } and { b$_{n}$ : n ≥ 1 } are two sequences of non-negative numbers, a$_{n}$ ≲ b$_{n}$ means there exists a finite positive C such that a$_{n}$ ≤ Cb$_{n}$ for all n sufficiently large, and a$_{n}$ ≍ b$_{n}$ means a$_{n}$ ≲ b$_{n}$ and b$_{n}$ ≲ a$_{n}$ . # S denotes the cardinality of a set S of finitely many elements. Given a strictly stationary process { X$_{i}$ } and 1 ≤ p < ∞ , we let L $^{p}$( X ) denote the function space consisting of all (equivalence classes) of measurable functions f for which the L $^{p}$( X ) norm ‖ f ‖ L $^{p}$( X ) ≡ E [ | f ( X$_{i}$ ) | $^{p}$] 1 /p is finite, and we let L $^{∞}$( X ) denote the space of bounded functions under the sup norm ‖ · ‖ $_{∞}$, i.e., if f : X → $^{R}$then ‖ f ‖ ∞ ≡ sup$_{x}$$_{∈X}$ | f ( x ) | . 

## 2 Uniform Convergence Rates 

In this section we present some general results on uniform convergence properties of nonparametric series LS estimators with weakly dependent data. 

## 2.1 Estimator and basic assumptions 

In nonparametric series LS estimation, the conditional mean function h$_{0}$ is estimated by least squares regression of Y$_{1}$, . . . , Y$_{n}$ on a vector of sieve basis functions evaluated at X$_{1}$, . . . , X$_{n}$ . The standard 

