bounded by uniformly for x$_{n}$ ∈ S $_{n}$. Moreover, ≲ ζ $_{K,n}$λ $_{K,n}$/n on A n (67) 1 n 2 n ∑ i =1 E [( g$_{i,n}$ϵ$_{1}$$_{,i,n}$ ) $^{2}$| F i − $_{1}$] ≲ n − 1 ˜ b K $_{w}$( x$_{n}$ ) ′ ( ˜ B ′ w ˜ B$_{w}$/n ) ˜ b K $_{w}$( x$_{n}$ ) (66) 2 2 | n − $^{1}$g$_{i,n}$ ϵ$_{1}$$_{,i,n}$ | ≲ ζ 2 $_{K,n}$λ 2 $_{K,n}$M$_{n}$ n (68) (# S $_{n}$) max x$_{n}$ ∈S$_{n}$ P ({∣ ∣ ∣$^{1}$ n n ∑ i =1 g$_{i,n}$ϵ$_{1}$$_{,i,n}$ ∣ > C 2 ζ$_{K,n}$λ$_{K,n}$ √ (log n ) /n } ∩ A n ) ≲ n ν$_{1}$ + η$_{2}$$^{ν$_{2}$}$exp { − C $^{2}$ζ2 $_{K,n}$λ 2 $_{K,n}$(log n ) /n c$_{1}$ζ 2 $_{K,n}$λ 2 $_{K,n}$/n + c$_{2}$ζ 2 $_{K,n}$λ 2 $_{K,n}$M$_{n}$ /n × Cζ$_{K,n}$λ$_{K,n}$ √ (log n ) /n } (69) C $^{2}$ζ2 $_{K,n}$λ 2 $_{K,n}$(log n ) /n $_{C}$√$_{n}$$_{log}$$_{n}$ ≲ exp { log n − C $^{2}$ζ2 $_{K,n}$λ 2 $_{K,n}$(log n ) /n c$_{3}$ζ 2 $_{K,n}$λ 2 $_{K,n}$/n } + exp { log n − $_{C}$√$_{n}$$_{log}$$_{n}$ c$_{4}$ζ$_{K,n}$λ$_{K,n}$M$_{n}$ } (70) uniformly for x$_{n}$ ∈ S $_{n}$. An tail bound for martingales (Freedman, 1975, Proposition 2.1) then provides that for finite positive constants c$_{1}$, . . . , c$_{4}$ . Thus (65a) vanishes asymptotically for all sufficiently large C provided M$_{n}$ = O ( ζ − 1 $_{K,n}$λ − 1 K,n √ n/ (log n )). Choosing M$_{n}$ as in the i.i.d. case completes the proof. Proof of Remark 2.5. Take any h ∈ L ∞ w,n with ‖ h ‖ ∞ ,w = 0. By the Cauchy-Schwarz inequality we have uniformly over x , where H = ( h ( X$_{1}$ ) w$_{n}$ ( X$_{1}$ ) , . . . , h ( X$_{n}$ ) w$_{n}$ ( X$_{n}$ )) $^{′}$. When λ$_{min}$ ( ˜ B ′ w ˜ B$_{w}$/n ) ≥ 1 2 (which it is wpa1 since ‖ ˜ B ′ w ˜ B$_{w}$/n − I$_{K}$ ‖ = o$_{p}$ (1)), we have: ‖ ( B ′ w B$_{w}$/n ) − ˜ B ′ $_{w}$H/n ‖ 2 = ( H ′ ˜ B$_{w}$/n )( ˜ B ′ w ˜ B$_{w}$/n ) − $^{1}$( ˜ B ′ w ˜ B$_{w}$/n ) − 1 ˜ B ′ $_{w}$H/n ≤ 2( H ′ B$_{w}$/n )( B ′ w B$_{w}$/n ) − 1 B ′ $_{w}$H/n ≤ 2 ‖ h ‖ 2 w,n ≤ 2 ‖ h ‖ 2 ∞ ,w 37 ≤ ‖ ˜ b K $_{w}$( x ) ‖‖ ( ˜ B ′ w ˜ B$_{w}$/n ) − ˜ B ′ $_{w}$H/n ‖ (71) ≤ ζ$_{K,n}$λ$_{K,n}$ ‖ ( B ′ w B$_{w}$/n ) − B ′ $_{w}$H/n ‖ (72) | P$_{K,w,n}$ ( x ) | ≤ ‖ b $_{w}$( x ) ‖‖ ( B w B$_{w}$/n ) B $_{w}$H/n ‖ (71)