regression with i.i.d. data, Huang (2003b) shows that the sup norm of the empirical L 2 projection onto splines is bounded with probability approaching one (wpa1). Using our new exponential inequality for sums of weakly dependent random matrices, his bound is easily extended to spline regression with weakly dependent regressors. In addition, we show in Theorem 5.2 that, for either i.i.d. or weakly dependent regressors, the sup norm of the empirical L 2 projection onto compactly supported wavelet bases is also bounded wpa1 (this property is called sup-norm stability of empirical L 2 projection). These tight bounds lead to sharp sup-norm bias control for spline and wavelet series LS estimators. They allow us to show that spline and wavelet series LS estimators achieve the optimal sup-norm convergence rate with weakly dependent data and heavy-tailed errors (e.g., E [ ϵ 4 $_{i}$] = ∞ is allowed). Sup-norm (uniform) convergence rates of series LS estimators have been studied previously by Newey (1997) and de Jong (2002) for i.i.d. data and Lee and Robinson (2013) for spatially dependent data. But the uniform convergence rates obtained in these papers are slower than the optimal rate of Stone (1982). 3 (2014) establish the attainability of the optimal sup-norm rates of series LS estimators using spline, local polynomial partition, wavelet and other series possessing the property of sup-norm stability of the L 2 projection (or bounded Lebesgue constant) under the conditional moment condition sup$_{x}$ E [ | ϵ$_{i}$ | 2+ δ | X$_{i}$ = x ] < ∞ for some δ > d/p . Our Theorem 5.1 on the sup-norm stability of L 2 projection of the wavelet basis is used by Belloni et al. (2014) to show that the wavelet series LS estimator achieves the op- timal sup-norm rate under their conditional moment requirement. Our paper contributes to the lit- erature by showing that spline and wavelet series LS estimators attain the optimal sup-norm rate with strictly stationary beta-mixing regressors under the weaker unconditional moment requirement E [ | ϵ$_{i}$ | 2+( d/p $^{)}$] < ∞ . As another application of our new exponential inequality, under very weak conditions we obtain sharp L 2 convergence rates for series LS estimators with weakly dependent regressors. For example, under the minimal bounded conditional second moment restriction (sup$_{x}$ E [ | ϵ$_{i}$ | 2 | X$_{i}$ = x ] < ∞ ), our L $^{2}$-norm rates for trigonometric polynomial, spline or wavelet series LS estimators attain Stone (1982)’s optimal L $^{2}$-norm rate of n − p/ (2 p + d ) with strictly stationary, exponentially beta-mixing (respectively algebraically beta-mixing at rate γ ) regressors with p > 0 (resp. p > d/ (2 γ )), while the power series LS $^{3}$See Hansen (2008), Kristensen (2009), Masry (1996), Tsybakov (2009), Cattaneo and Farrell (2013) and the references therein for attainability of the optimal uniform convergence rates with kernel, local linear regression and partitioning estimators. 3