content_image/311844.jpg content_image/311845.jpg content_image/311843.jpg content_image/311846.jpg content_image/311847.jpg content_image/311851.jpg content_image/311854.jpg content_image/311853.jpg
bounded by 

uniformly for x$_{n}$ ∈ S $_{n}$. Moreover, 

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uniformly for x$_{n}$ ∈ S $_{n}$. An tail bound for martingales (Freedman, 1975, Proposition 2.1) then provides that 

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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: ′ − ′ 2 ′ ′ − ′ − 1 ′ 

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