1 Introduction We study the nonparametric regression model where Y$_{i}$ ∈ R is a scalar response variable, X$_{i}$ ∈ X ⊆ R d is a d -dimensional regressor (predictor variable), and the conditional mean function h$_{0}$ ( x ) = E [ Y$_{i}$ | X$_{i}$ = x ] belongs to a H¨ older space of smoothness p > 0. We are interested in series least squares (LS) estimation 1 of h$_{0}$ under sup-norm loss and inference on possibly nonlinear functionals of h$_{0}$ allowing for weakly dependent regressors and heavy-tailed errors ϵ$_{i}$ . 2 For i.i.d. data, Stone (1982) shows that ( n/ log n ) − p/ (2 p + d ) is the minimax lower bound in sup-norm risk for estimation of h$_{0}$ over a H¨ older ball of smoothness p > 0. For strictly stationary beta-mixing regressors, we show that spline and wavelet series LS estimators ̂ h of h$_{0}$ attain the optimal uniform rate of Stone (1982) under a mild unconditional moment condition E [ | ϵ$_{i}$ | 2+( d/p $^{)}$] < ∞ imposed on the martingale difference errors. More generally, we assume the error process { ϵ$_{i}$ } ∞ i = −∞ is a martingale difference sequence but impose no explicit weak dependence condition on the regressor process { X$_{i}$ } ∞ i = $_{−∞}$. Rather, weak de- pendence of the regressor process is formulated in terms of convergence of a certain random matrix. We verify this condition for absolutely regular (beta-mixing) sequences by deriving a new exponential inequality for sums of weakly dependent random matrices. This new inequality then leads to a sharp upper bound on the sup-norm variance term of series LS estimators with an arbitrary basis. When combined with a general upper bound on the sup-norm bias term of series LS estimators, the sharp sup-norm variance bound immediately leads to a general upper bound on the sup-norm convergence rate of series LS estimators with an arbitrary basis and weakly dependent data. In our sup-norm bias and variance decomposition of series LS estimators, the bound on the sup- norm bias term depends on the sup norm of the empirical L 2 projection onto the linear sieve space. The sup norm of the empirical L 2 projection varies with the choice of the (linear sieve) basis. For spline $^{1}$Other terms for series LS appearing in the literature include series regression and linear sieve regression, but we use series LS hereafter. The series LS estimator falls into the general class of nonparametric sieve M-estimators. $^{2}$The error ϵ$_{i}$ is heavy-tailed in the sense that E [ | ϵ$_{i}$ | 2+ $^{δ}$] = ∞ for δ > d/p is allowed; say E [ | ϵ$_{i}$ | $^{4}$] = ∞ is allowed. 2 Y$_{i}$ = h$_{0}$ ( X$_{i}$ ) + ϵ$_{i}$ E [ ϵ$_{i}$ | X$_{i}$ ] = 0 (1)