required. Theorem 7.1 Under conditions stated in Theorem 5.1, we have ‖ P$_{K,n}$ ‖ ∞ ≲ 1 wpa1 provided the following are satisfied: (i) ‖ ( B $^{′}$B/n ) − E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ‖ = o$_{p}$ (1) , and (ii) max$_{1}$$_{≤}$$_{k}$$_{≤}$$_{K}$ ∣ 1 n ∑ n i $_{=1}$| b$_{Kk}$ ( X$_{i}$ ) |− E [ | b$_{Kk}$ ( X$_{i}$ ) | ] E [ | b$_{Kk}$ ( X$_{i}$ ) | ] ∣ = o$_{p}$ (1) . Proof of Theorem 7.1. Condition (ii) max$_{1}$$_{≤}$$_{k}$$_{≤}$$_{K}$ 1 n ∑ n i $_{=1}$| b$_{Kk}$ ( X$_{i}$ ) |− E [ | b$_{Kk}$ ( X$_{i}$ ) | ] E [ | b$_{Kk}$ ( X$_{i}$ ) | ] = o$_{p}$ (1) implies max 1 ≤ k ≤ K 1 n n ∑ i =1 | b$_{Kk}$ ( X$_{i}$ ) | ≲ max 1 ≤ k ≤ K ‖ b$_{Kk}$ ‖ L $^{1}$( X ) ≲ K − 1 / 2 (135) where the final inequality is by the proof of Theorem 5.1. Moreover, sup$_{x}$ ‖ b $^{K}$( x ) ‖ ℓ 1 ≲ √$_{K}$ by the proof of Theorem 5.1. It follows analogously to Lemma 7.1 that ‖ P$_{K,n}$ ‖ ∞ ≲ ‖ ( B $^{′}$B/n ) − 1 ‖ ∞ wpa1 (noting that B $^{′}$B/n is invertible wpa1 because ‖ ( B $^{′}$B/n ) − E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ‖ = o$_{p}$ (1) and λ$_{K,n}$ ≲ 1). Condition (i) ‖ ( B $^{′}$B/n ) − E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ‖ = o$_{p}$ (1) implies (1) λ$_{min}$ ( B $^{′}$B/n ) ≳ λ$_{min}$ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$]), (2) λ$_{max}$ ( B $^{′}$B/n ) ≲ λ$_{max}$ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$]), and (3) ‖ ( B $^{′}$B/n ) − 1 ‖ ≲ ‖ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ) − 1 ‖ all hold wpa1. Moreover, λ$_{min}$ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$]) ≳ 1 and λ$_{max}$ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$]) ≲ 1 by the proof of Theorem 5.1. It follows by Lemma 7.2 that ‖ ( B $^{′}$B/n ) − 1 ‖ ℓ ∞ ≲ 1 wpa1, as required. Proof of Theorem 5.2. Condition (i) of Theorem 7.1 is satisfied because λ$_{K,n}$ ≲ 1 and the condition ‖ ( ˜ B ′ ˜ B/n ) − I$_{K}$ ‖ = o$_{p}$ (1) under the conditions on K (see Lemma 2.1 for the i.i.d. case and Lemma 2.2 for the weakly dependent case). Therefore, It remains to verify condition (ii) of Theorem 7.1. Let b$_{K}$$_{1}$ = ϕ d J, $_{0}$,... ,b$_{KK}$ = ϕ d J, 2 $^{J}$− 1 with K = 2 dJ as in the proof of Theorem 5.1. Similar arguments to the proof of Theorem 5.1 yield the bounds ‖ b$_{Kk}$ ‖ ∞ ≲ 2 dJ/ 2 = √$_{K}$ uniformly for 1 ≤ k ≤ K . Let f$_{X}$ ( x ) denote the density of X . Then by 50 ≤ [ λ$_{min}$ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$])]− $^{1}$‖ ( ˜ B ′ ˜ B/n ) − I$_{K}$ ‖ (136) ≲ ‖ ( ˜ B ′ ˜ B/n ) − I$_{K}$ ‖ = o$_{p}$ (1) . (137) ‖ ( B $^{′}$B/n ) − E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ‖ ≤ [ λ$_{min}$ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$])]− $^{1}$‖ ( B B/n ) − I$_{K}$ ‖ (136)