content_image/321775.jpg content_image/321778.jpg content_image/321777.jpg content_image/321788.jpg content_image/321781.jpg content_image/321782.jpg content_image/321783.jpg content_image/321784.jpg
triangle inequality we bound 

Recall the definition of the empirical projection P$_{K,w,n}$ from expression (15), and observe that ˜ h = P$_{K,w,n}$h$_{0}$ and that P$_{K,w,n}$h = h for all h ∈ B$_{K,w}$ . Also recall the definition of L 2 $_{w,n}$( X ) as the space of functions with finite norm ‖ · ‖ w,n where ‖ f ‖ 2 w,n = 1 n ∑ n i $_{=1}$f ( X$_{i}$ ) $^{2}$w$_{n}$ ( X$_{i}$ ). Since the empirical and theoretical L $^{2}$( X ) norms are equivalent over B$_{K,w}$ wpa1 under the condition ‖ ˜ B ′ w ˜ B$_{w}$/n − I$_{K}$ ‖ = o$_{p}$ (1) (see Section 4.2). Therefore, we have 

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where the second line is by equivalence of the empirical and theoretical L $^{2}$( X ) norms wpa1, and the final line is because P$_{K,w,n}$ is an orthogonal projection on L 2 $_{w,n}$( X ). Finally, Markov’s inequality yields ‖ ( h$_{0}$ − h$_{0}$$_{,K}$ ) ‖ 2 w,n = O$_{p}$ ( ‖ h$_{0}$ − h$_{0}$$_{,K}$ ‖ 2 L $^{2}$( X $_{)}$). 

## 7.2 Proofs for Section 3 

Proof of Lemma 3.1. We use a truncation argument together with exponential inequalities for random matrices. Let M$_{n}$ ≍ ( ζ$_{K,n}$λ$_{K,n}$ ) (2+ δ ) /δ (with δ as in Assumption 2(iii) be a sequence of positive numbers and let 

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Clearly ̂ Ω − Ω = ̂ Ω$_{1}$ + ̂ Ω$_{2}$, so it is enough to show that ‖ ̂ Ω$_{1}$ ‖ = o$_{p}$ (1) and ‖ ̂ Ω$_{2}$ ‖ = o$_{p}$ (1). Control of ‖ Ω$_{1}$ ‖ : By definition, ‖ Ξ$_{1}$$_{,i}$ ‖ ≤ M 2 $_{n}$. It follows by the triangle inequality and Jensen’s 

