for any h ∈ L $^{2}$( X ), and define its L ∞ operator norm: Under i.i.d. data, Assumptions 1, 2(i)(ii) (i.e., E [ ϵ$_{i}$ | X$_{i}$ ] = 0 , sup$_{x}$ E [ | ϵ$_{i}$ | 2 | X$_{i}$ = x ] < ∞ ), 6, and the conditions λ$_{K,n}$ ≲ 1, ζ 2 $_{K,n}$K/n = o (1) and ‖ h$_{0}$ − h ∗ 0 ,K ‖ ∞ ≲ K − p/d on the series basis, Newey (1997) derived the following sup-norm convergence rates for series LS estimators with an arbitrary basis: By Remark 2.5, under the same set of mild conditions imposed in Newey (1997) (except allowing for weakly dependent regressors), the bound (18) can be slightly improved to It is clear that the general bounds in (18) and (19) with an arbitrary basis are not optimal, but they are derived under the minimal moment restriction of Assumption 2(ii) without the existence of higher-than-second moments (i.e., δ = 0 in Assumption 2(iii)). Under the extra moment condition sup$_{x}$ E [ | ϵ$_{i}$ | 4 | X$_{i}$ = x ] < ∞ , de Jong (2002) obtained the follow- ing general bound on sup-norm rates for series LS estimators with an arbitrary basis: de Jong (2002) did not provide sharp bounds for ‖ P$_{K}$h$_{0}$ − h ∗ 0 ,K ‖ ∞ for any particular basis, and was therefore unable to attain the optimal convergence rate ‖ ̂ h − h$_{0}$ ‖ ∞ = O$_{p}$ (( n/ log n ) − p/ (2 p + d $^{)}$) of Stone (1982). Note that Given the newly derived sharp bounds of ‖ P$_{K}$ ‖ ∞ ≲ 1 in Huang (2003b) for splines, in Belloni et al. (2014) for the local polynomial partition series, and in our paper (Theorem 5.1) for wavelets, one could now apply de Jong (2002)’s result (20) to conclude the attainability of the optimal sup-norm ‖ ̂ h − h$_{0}$ ‖ ∞ ≤ O$_{p}$ ( ζ$_{K,n}$ √ (log n ) /n + K − p/d + ‖ P$_{K}$h$_{0}$ − h ∗ 0 ,K ‖ ∞ ) . (20) ‖ P$_{K}$h$_{0}$ − h ∗ 0 ,K ‖ ∞ = ‖ P$_{K}$ ( h$_{0}$ − h ∗ 0 $_{,K}$) ‖ ∞ ≤ ‖ P$_{K}$ ‖ ∞ ‖ h$_{0}$ − h ∗ 0 ,K ‖ ∞ ≲ ‖ P$_{K}$ ‖ $_{∞}$K − $^{p/d}$. (21) ‖ P$_{K}$ ‖ ∞ := sup h ∈ L $^{∞}$( X ): ‖ h ‖$_{∞}$̸ =0 ‖ P$_{K}$h ‖ ∞ ‖ h ‖ ∞ . (17) ‖ ̂ h − h$_{0}$ ‖ ∞ ≤ O$_{p}$ ( ζ$_{K,n}$ √ K/n + ζ$_{K,n}$K − p/d ) . (18) ‖ ̂ h − h$_{0}$ ‖ ∞ ≤ O$_{p}$ ( ζ$_{K,n}$ √ K/n + ‖ P$_{K,w,n}$ ‖ ∞ $_{,w}$K − p/d ) . (19) 15