estimator attains the same optimal rate with exponentially (resp. algebraically) beta-mixing regressors for p > d/ 2 (resp. p > d (2 + γ ) / (2 γ )). It is interesting to note that for a smooth conditional mean function, we obtain the optimal L 2 convergence rates for these commonly used series LS estimators with weakly dependent regressors without requiring the existence of higher-than-second unconditional moments of the error terms. Previously, Newey (1997) derived the optimal L 2 convergence rates of series LS estimator under i.i.d. data and the restriction of K $^{2}$/n = o (1) for spline and trigonometric series (and K $^{3}$/n = o (1) for power series), where K is the series number of terms. The restriction on K is relaxed to K (log K ) /n = o (1) in Huang (2003a) for splines and in Belloni et al. (2014) for wavelets, trigonometric and other series under i.i.d. data. We show that the optimal L 2 convergence rates are still attainable for splines, wavelets, trigonometric and other series under exponentially beta-mixing and K (log K ) $^{2}$/n = o (1). $^{√$_{n}$$_{(}$$_{f}$$_{(}$}$̂ h ) − f ( h$_{0}$ )) V 1 / 2 K → d N (0 , 1) We also show that feasible asymptotic inference can be performed on a possibly nonlinear functional f ( h$_{0}$ ) using the plug-in series LS estimator f ( ̂ h ). We establish the asymptotic normality of f ( ̂ h ) and of the corresponding Student t statistic for weakly dependent data under mild low-level conditions. When specializing to general irregular (i.e., slower than √$_{n}$$_{-estimable) but sup-norm bounded}$ linear functionals of spline or wavelet series LS estimators with i.i.d. data, we obtain the asymptotic normality of f ( ̂ h ) h ) − f ( h$_{0}$ )) 1 / 2 d N (0 , 1) under remarkably mild conditions of (1) uniform integrability (sup$_{x}$$_{∈X}$ E [ ϵ 2 i {| ϵ$_{i}$ | > ℓ ( n ) }| X$_{i}$ = x ] → 0 for any ℓ ( n ) → ∞ as n → ∞ ), and (2) K − p/d √ n/V$_{K}$ = o (1), ( K log K ) /n = o (1), where V$_{K}$ is the sieve variance that grows with K for irregular functionals. These conditions coincide with the weakest known conditions in Huang (2003b) for the pointwise asymptotic normality of spline LS estimators, except we also allow for other irregular linear functionals of spline or wavelet LS estimators. 4 When specializing to general irregular but sup-norm bounded nonlinear functionals of spline or wavelet series LS estimators with i.i.d. data, we obtain asymptotic normality of f ( ̂ h ) (and of its t statistic) under conditions (1) and (3) K − p/d √ n/V$_{K}$ = o (1), K (2+ δ ) $^{/δ}$(log n ) /n ≲ 1 (and K (2+ δ ) $^{/δ}$(log n ) /n = o (1) for the t statistic) for δ ∈ (0 , 2) such that E [ | ϵ$_{i}$ | 2+ $^{δ}$] < ∞ . These conditions are much weaker than the well-known conditions in Newey (1997) for the asymptotic normality of a nonlinear functional and its t $^{4}$Most of the literature has focused on the case of i.i.d. or strictly stationary data. See Andrews (1991) for asymptotic normality of linear functionals of series LS estimators with non-identically distributed data. 4