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## 1 Introduction 

We study the nonparametric regression model 

<img src='content_image/318232.jpg'>

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 

