Assumption 3 is trivially satisfied when X is compact and D n = X for all n . More generally, when X is noncompact and D n is an expanding sequence of compact subsets of X as described above, Assumption 3(ii) is satisfied provided each D n is contained in an Euclidean ball of radius r$_{n}$ ≲ n ν for some ν > 0. 6 Assumptions 1 and 2 trivially nest i.i.d. sequences, but also allow the regressors to exhibit quite general weak dependence. Note that Assumption 2(ii) reduces to sup$_{x}$ E [ ϵ 2 i | X$_{i}$ = x ] < ∞ in the i.i.d. case. Suitable choice of δ in Assumption 2(iii) for attainability of the optimal uniform rate will be explained subsequently. Strict stationarity of { ϵ$_{i}$ } in Assumption 2 may be dropped provided the sequence {| ϵ$_{i}$ | 2+ $^{δ}$} is uniformly integrable. However, strict stationarity is used to present simple sufficient conditions for the asymptotic normality of functionals of ̂ h in Section 3. Assumption 3 is trivially satisfied when is compact and n = for all n . More generally, Assumption 4 is a mild regularity condition on the sieve basis functions. When X is compact and rectangular this assumption is satisfied by all the widely used series (or linear sieve bases) with λ$_{K,n}$ ≲ 1, and ζ$_{K,n}$ ≲ √$_{K}$ for tensor-products of univariate polynomial spline, trigonometric polynomial or wavelet bases, and ζ$_{K,n}$ ≲ K for tensor-products of power series or orthogonal polynomial bases (see, Assumption 2 (i) { ϵ$_{i}$, F i − 1 } n i =1 with F i − 1 = σ ( X$_{i}$, ϵ$_{i}$$_{−}$$_{1}$, X$_{i}$$_{−}$$_{1}$, . . . ) is a strictly stationary martingale difference sequence, (ii) E [ ϵ 2 i |F i − $_{1}$] is uniformly bounded for all i ≥ 1 , almost surely, (iii) E [ | ϵ$_{i}$ | 2+ $^{δ}$] < ∞ for some δ > 0 . Let N ( D $_{n}$, ϵ ) denote the internal ϵ -covering number of D n with respect to the Euclidean norm (i.e. the minimum number of points x$_{1}$, . . . , x$_{m}$ ∈ D n such that the collection of ϵ -balls centered at each of x$_{1}$, . . . , x$_{m}$ cover D $_{n}$). Assumption 3 (i) D n is compact, convex, has nonempty interior, and D n ⊆ D n +1 for all n , (ii) there exists ν$_{1}$, ν$_{2}$ > 0 such that N ( D $_{n}$, ϵ ) ≲ n $^{ν$_{1}$}$ϵ − $^{ν$_{2}$}$. Define ζ$_{K,n}$ ≡ sup$_{x}$ ‖ b K $_{w}$( x ) ‖ and λ$_{K,n}$ ≡ [ λ$_{min}$ ( E [ b K $_{w}$( X$_{i}$ ) b K $_{w}$( X$_{i}$ ) $^{′}$]) ] − 1 / $^{2}$. Assumption 4 (i) there exist ω$_{1}$, ω$_{2}$ ≥ 0 s.t. sup$_{x}$$_{∈D}$$_{n}$ ‖∇ b K $_{w}$( x ) ‖ ≲ n $^{ω$_{1}$}$K$^{ω$_{2}$}$, (ii) there exist ϖ$_{1}$ ≥ 0 , ϖ$_{2}$ > 0 s.t. ζ$_{K,n}$ ≲ n $^{ϖ$_{1}$}$K$^{ϖ$_{2}$}$, (iii) λ$_{min}$ ( E [ b K $_{w}$( X$_{i}$ ) b K $_{w}$( X$_{i}$ ) $^{′}$]) > 0 for each K and n . $^{6}$By translational invariance we may assume that D n is centered at the origin. Then D n ⊆ R n = [ − r$_{n}$, r$_{n}$ ] $^{d}$. We can cover R n with ( r$_{n}$/ϵ ) d ℓ $^{∞}$-balls of radius ϵ , each of which is contained in an Euclidean ball of radius $_{ϵ}$√$_{d}$$_{. Therefore,}$ N ( D $_{n}$, ϵ ) ≤ $_{(}$√$_{dr}$$_{n}$$_{)}$ $^{d}$ϵ − d ≲ n $^{νd}$ϵ − $^{d}$. 9