Control of (58): The Cauchy-Schwarz inequality and Assumption 4(iii) yield | ˜ b K $_{w}$( x$_{n}$ ) $^{′}${ ( ˜ B ′ w ˜ B$_{w}$/n ) − − I$_{K}$ } ˜ B ′ $_{w}$e/n | ≲ ζ$_{K,n}$λ$_{K,n}$ ‖ ( ˜ B ′ w ˜ B$_{w}$/n ) − − I$_{K}$ ‖ × O$_{p}$ ( √ K/n ) (60) ‖ ( ˜ B ′ w ˜ B$_{w}$/n ) − − I$_{K}$ ‖ = ‖ ( ˜ B ′ w ˜ B$_{w}$/n ) − $^{1}$(( ˜ B ′ w ˜ B$_{w}$/n ) − I$_{K}$ ) ‖ ≤ 2 ‖ ( ˜ B ′ w ˜ B$_{w}$/n ) − I$_{K}$ ‖ . (61) uniformly for x$_{n}$ ∈ S n (since ‖ ˜ B ′ $_{w}$e/n ‖ = O$_{p}$ ( √ K/n ) under Assumption 2(i)(ii)). On A n we have Thus ‖ ˜ B ′ w ˜ B$_{w}$/n − I$_{K}$ ‖ = O$_{p}$ ( √ (log n ) /K ) (i.e. condition (ii)) ensures that (58) can be made arbitrarily small for large enough C . Control of (59): Let M$_{n}$ be as in the i.i.d. case and define ϵ$_{1}$$_{,i,n}$ := ϵ$_{i}$ {| ϵ$_{i}$ | ≤ M$_{n}$ } − E [ ϵ$_{i}$ {| ϵ$_{i}$ | ≤ M$_{n}$ }|F i − $_{1}$] (62) ϵ$_{2}$$_{,i,n}$ := ϵ$_{i}$ − ϵ$_{1}$$_{,i,n}$ (63) g$_{i,n}$ ( x$_{n}$ ) := ˜ b K $_{w}$( x$_{n}$ ) ′ ˜ b K $_{w}$( X$_{i}$ ) {A n } . (64) The relation $^{P}$( A ) ≤ $^{P}$( A ∩ B ) + $^{P}$( B $^{c}$) and the triangle inequality together yield P ( max x$_{n}$ ∈S$_{n}$ | ˜ b K $_{w}$( x$_{n}$ ) ′ ˜ B ′ $_{w}$e/n | ≥ Cζ$_{K,n}$λ$_{K,n}$ √ (log n ) /n ) − $^{P}$( A c $_{n}$) ≤ (# S $_{n}$) max x$_{n}$ ∈S$_{n}$ P ({∣ ∣ ∣$^{1}$ n n ∑ i =1 g$_{i,n}$ϵ$_{1}$$_{,i,n}$ ∣ > C 2 ζ$_{K,n}$λ$_{K,n}$ √ (log n ) /n } ∩ A n ) (65a) P n C + P ( max x$_{n}$ ∈S$_{n}$ ∣ ∣$^{1}$ n n ∑ i =1 g$_{i,n}$ϵ$_{2}$$_{,i,n}$ ∣ ≥ C 2 ζ$_{K,n}$λ$_{K,n}$ √ (log n ) /n ) . (65b) Control of (65b): First note that | g$_{i,n}$ | ≤ ζ 2 $_{K,n}$λ 2 K,n by the Cauchy-Schwarz inequality and Assump- tion 4(iii). This, together with Markov’s inequality and Assumption 2(iii) yields P ( max x$_{n}$ ∈S$_{n}$ ∣ ∣$^{1}$ n n ∑ i =1 g$_{i,n}$ϵ$_{2}$$_{,i,n}$ ∣ ≥ C 2 ζ$_{K,n}$λ$_{K,n}$ √ (log n ) /n ) ≲ ζ 2 $_{K,n}$λ 2 $_{K,n}$E [ | ϵ$_{i}$ |{| ϵ$_{i}$ | > M$_{n}$ } ] ζ$_{K,n}$λ$_{K,n}$ √ (log n ) /n ≤ ζ$_{K,n}$$_{λ$_{K,n}$}$√$_{n}$ √$_{log}$$_{n}$ E [ | ϵ$_{i}$ | 2+ $^{δ}${ | ϵ$_{i}$ | > M$_{n}$ } ] M 1+ δ n which is o (1) provided ζ$_{K,n}$λ$_{K,n}$ n/ log n = O ( M 1+ δ n ). Control of (65a): By Assumption 2(ii), the predictable variation of the summands in (65a) may be 36