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(a) { ( X$_{i}$, Y$_{i}$ ) } n i =1 is i.i.d.; 

(b) { X$_{i}$ } n i =1 is weakly dependent: Assumption 8 holds, and ‖ ˜ B ′ w ˜ B$_{w}$/n − I$_{K}$ ‖ = o$_{p}$ ( K − 1 / $^{2}$) . 

We now consider the special case of irregular but sup-norm bounded linear or nonlinear functionals as discussed in Remark 3.1. The sup-norm convergence rates for series LS estimators in Section 2 are employed to derive asymptotic normality of plug-in estimators of such functionals under weak conditions. To save space, for the weakly dependent case we only present sufficient conditions for asymptotic normality of f ( ̂ h ) when the regression error has no more than a finite 4th absolute moment (i.e., E [ | ϵ$_{i}$ | 2+ $^{δ}$] < ∞ for some 0 ≤ δ ≤ 2). We also take X = [0 , 1] d and w$_{n}$ = 1 for all n for simplicity. 

Corollary 3.1 Let f be an irregular but sup-norm bounded linear functional, and let Assumptions 1 (with X = [0 , 1] $^{d}$), 2(i)(ii)(iv)(v), 6, and 7 hold. Then 

as n, K → ∞ provided that either (a), (b) or (c) is satisfied: 

(a) { ( X$_{i}$, Y$_{i}$ ) } n i =1 is i.i.d.: √$_{nV}$ − 1 / 2 K K − p/d = o (1) and ( K log K ) /n = o (1) ; 

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(b) { ( X$_{i}$, Y$_{i}$ ) } n i =1 is exponentially β -mixing: Assumption 2(iii) also holds, √$_{nV}$ − 1 / 2 K K − p/d = o (1) , and K (2+ δ ) $^{/δ}$(log n ) $^{2}$/n = o (1) with δ ≤ 2 ; 

(c) { ( X$_{i}$, Y$_{i}$ ) } n i =1 is algebraically β -mixing at rate γ : Assumption 2(iii) also holds, √$_{nV}$ − 1 / 2 K K − p/d = o (1) , and K (2+ δ ) $^{/δ}$(log K ) /n γ/ (1+ γ ) = o (1) with δ ≤ 2 . 

Corollary 3.1 part (a) extends the weakest known result on pointwise asymptotic normality of spline LS estimators in Huang (2003b) to general sup-norm bounded linear functionals of spline or wavelet series LS estimators. 9 

Corollary 3.2 Let f be an irregular but sup-norm bounded nonlinear functional, and let Assumptions 1 (with X = [0 , 1] $^{d}$), 2, 6, 7 and 9(i’)(ii’)(iii’) hold. Then 

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