content_image/297424.jpg content_image/297426.jpg
Lemma 2.2 Under Assumption 4(iii), if { X$_{i}$ } ∞ i = −∞ is strictly stationary and β -mixing with mixing coefficients such that one can choose an integers q = q ( n ) ≤ n/ 2 with β ( q ) n/q = o (1) , then 

provided ζ$_{K,n}$λ$_{K,n}$ √ ( q log K ) /n = o (1) . 

<img src='content_image/297424.jpg'>

Remark 2.3 Consider the compact support case from Remark 2.2, with { X$_{i}$ } ∞ i = −∞ strictly stationary and β -mixing. 

(i) Exponential β -mixing: Assumption 5(b) is satisfied with √ K (log n ) $^{2}$/n = o (1) for (tensor- product) spline, trigonometric polynomial or wavelet bases, and with K √ (log n ) $^{2}$/n = o (1) for (tensor-product) power series. 

(ii) Algebraic β -mixing at rate γ : Assumption 5(b) is satisfied with √ ( K log K ) /n γ/ (1+ γ ) = o (1) for (tensor-product) spline, trigonometric polynomial or wavelet bases, and with K √ (log K ) /n γ/ (1+ γ ) = o (1) for (tensor-product) power series. 

## 2.2 A general upper bound on uniform convergence rates 

Let B$_{K,w}$ = clsp { b$_{K}$$_{1}$w$_{n}$, . . . , b$_{KK}$w$_{n}$ } be a general weighted linear sieve space. Let ˜ h denote the projection of h$_{0}$ onto B$_{K,w}$ under the empirical measure, that is, 

where H$_{0}$ = ( h$_{0}$ ( X$_{1}$ ) , . . . , h$_{0}$ ( X$_{n}$ )) $^{′}$. The sup-norm distance ‖ ̂ h − h$_{0}$ ‖ ∞ ,w may be trivially bounded using 

<img src='content_image/297426.jpg'>

Sharp bound on the sup-norm variance term . The following result establishes a sharp uniform convergence rate of the variance term for an arbitrary linear sieve space. Convergence is established 

