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Let Ξ ∗ i,n = Ξ( X ∗ $_{i,n}$). By the triangle inequality, 

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To control the last two terms we apply Theorem 4.1, recognizing that i ∈ I$_{e}$ Ξ ∗ i,n and ∑ i ∈ I$_{o}$ Ξ ∗ i,n are each the sum of fewer than [ n/q ] independent d$_{1}$ × d$_{2}$ matrices, namely W ∗ k = ∑ kq i =( k − 1) q $_{+1}$Ξ ∗ $_{i,n}$. Moreover each W ∗ k satisfies ‖ W ∗ k ‖ ≤ qR$_{n}$ and max {‖ E [ W ∗ $_{k}$W ∗′ k ] ‖ , ‖ E [ W ∗′ k W ∗ $_{k}$] ‖} ≤ q $^{2}$s$_{n}$ . Theorem 4.1 then yields 

and similarly for I$_{o}$ . 

Proof of Corollary 4.2. Follows from Theorem 4.2 with t = Cs$_{n}$ √ nq log( d$_{1}$ + d$_{2}$ ) for sufficiently large C , and the conditions n $_{q}$β ( q ) = o (1) and R$_{n}$ √ q log( d$_{1}$ + d$_{2}$ ) = o ( $_{s$_{n}$}$√$_{n}$$_{).}$ Proof of Lemma 4.1. Let G = E [ b K $_{w}$( X$_{i}$ ) b K $_{w}$( X$_{i}$ ) $^{′}$]. Since B$_{K,w}$ = clsp b$_{K}$$_{1}$w$_{n}$, . . . , b$_{KK}$w$_{n}$ , we have: 

{ } 

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as required. 

## 7.4 Proofs for Section 5 

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We first present a general result that allows us to bound the L ∞ operator norm of the L $^{2}$( X ) pro- jection P$_{K}$ onto a linear sieve space B$_{K}$ ≡ clsp { b$_{K}$$_{1}$, . . . , b$_{KK}$ } by the ℓ ∞ norm of the inverse of its corresponding Gram matrix. 

Lemma 7.1 If there exists a sequence of positive constants { c$_{K}$ } such that (i) sup$_{x}$$_{∈X}$ ‖ b $^{K}$( x ) ‖ ℓ 1 ≲ c$_{K}$ 

