higher-than-second conditional moment. Thanks to our Lemma 2.2, we are able to show that series LS estimators with arbitrary bases attain the optimal L 2 convergence rate with beta-mixing regressors under a uniformly bounded second conditional moment condition on the residuals. Our result should be very useful to nonparametric series regression for financial time series data with heavy-tailed errors. We now study inference on possibly nonlinear functionals f : L $^{2}$( X ) ∩ L $^{∞}$( X ) → R of the regres- sion function h$_{0}$ . Examples of functionals include, but are not limited to, the pointwise evaluation functional, the partial mean functional, and consumer surplus (see, e.g., Newey (1997) for examples). The functional f ( h$_{0}$ ) may be estimated using the plug-in series LS estimator f ( ̂ h ), for which we now establish feasible limit theory. As with Newey (1997) and Chen et al. (2014), our results allow researchers to perform inference on nonlinear functionals f of h$_{0}$ without needing to know whether or not f ( h$_{0}$ ) is regular (i.e., √$_{n}$$_{-}$ estimable). However, there is already a large literature on the √$_{n}$$_{-asymptotic normality and the}$ consistent variance estimation for series estimators of regular functionals of conditional mean functions with weakly dependent data (see, e.g., Chen and Shen (1998), Chen (2007), Li and Racine (2006)). To save space and to illustrate the usefulness of our new sup-norm convergence rate results, we focus on asymptotic normality of f ( ̂ h ) and the corresponding sieve t statistic when the functional is irregular (i.e., slower than √$_{n}$$_{-estimable) in this section.}$ We borrow some notation and definitions from Chen et al. (2014). Denote the pathwise derivative of f at h$_{0}$ in the direction v ∈ V := ( L $^{2}$( X ) − { h$_{0}$ } ) by and assume it is linear. Let v ∗ K ∈ V K := ( B$_{K,w}$ − { h$_{0}$$_{,K}$ } ) be the sieve Riesz representer of ∂f ( h$_{0}$ ) ∂h [ · ] on V$_{K}$ , i.e. v ∗ K is the unique element of V K such that ∂f ( h$_{0}$ ) ∂h [ v ] := lim τ → 0 + f ( h$_{0}$ + τv ) τ (23) ∂f ( h$_{0}$ ) ∂h [ v ] = E [ v ∗ $_{K}$( X$_{i}$ ) v ( X$_{i}$ )] for all v ∈ V $_{K}$. (24) 18 3 Inference on possibly nonlinear functionals