Let Ξ ∗ i,n = Ξ( X ∗ $_{i,n}$). By the triangle inequality, $^{P}$( ‖ ∑ n i $_{=1}$Ξ$_{i,n}$ ‖ ≥ 6 t ) ≤ $^{P}$( ‖ [ n/q ] q i =1 Ξ ∗ i,n ‖ + ‖ ∑ i ∈ I$_{r}$ Ξ$_{i,n}$ ‖ + ‖ ∑ [ n/q ] q i =1 (Ξ ∗ i,n − Ξ$_{i,n}$) ‖ ≥ 6 t ) ≤ n $_{q}$β ( q ) + P ( ‖ ∑ i ∈ I$_{r}$ Ξ$_{i,n}$ ‖ ≥ t ) + P ( ‖ i ∈ I$_{e}$ Ξ ∗ i,n ‖ ≥ t ) + P ( ‖ ∑ i ∈ I$_{o}$ Ξ ∗ i,n ‖ ≥ t ) (118) 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 P (∥ ∥ ∑ i ∈ I$_{e}$ Ξ ∗ i,n ∥ ≥ t ) ≤ ( d$_{1}$ + d$_{2}$ )exp ( − t $^{2}$/ 2 nqs 2 n + qR$_{n}$t/ 3 ) (119) 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}$$_{).}$ K K Proof of Lemma 4.1. Let G = E [ b $_{w}$( X$_{i}$ ) b $_{w}$( X$_{i}$ ) $^{′}$]. Since B$_{K,w}$ = clsp { b$_{K}$$_{1}$w$_{n}$, . . . , b$_{KK}$w$_{n}$ } , we have: as required. 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$_{K}$ 7.4 Proofs for Section 5 sup {| 1 n ∑ n i $_{=1}$b ( X$_{i}$ ) 2 − 1 | : b ∈ B$_{K,w}$, E [ b ( X ) $^{2}$] = 1 } = sup {| c $^{′}$( B ′ $_{w}$B$_{w}$ /n − G ) c | : c ∈ R $^{K}$, ‖ G 1 / $^{2}$c ‖ = 1 } (120) = sup {| c $^{′}$G 1 / $^{2}$( G − 1 / $^{2}$( B ′ $_{w}$B$_{w}$ /n ) G − 1 / 2 − I$_{K}$ ) G 1 / $^{2}$c | : c ∈ R $^{K}$, ‖ G 1 / $^{2}$c ‖ = 1 } (121) = sup c $^{′}$( B B$_{w}$/n I$_{K}$ ) c : c $^{K}$, c = 1 (122) = sup {| c $^{′}$( ˜ B ′ w ˜ B$_{w}$/n − I$_{K}$ ) c | : c ∈ R $^{K}$, ‖ c ‖ = 1 } (122) = ‖ ˜ B ′ w ˜ B$_{w}$/n − I$_{K}$ ‖ 2 (123) 46