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Control of (58): The Cauchy-Schwarz inequality and Assumption 4(iii) yield 

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 

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The relation $^{P}$( A ) ≤ $^{P}$( A ∩ B ) + $^{P}$( B $^{c}$) and the triangle inequality together yield 

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 

which is o (1) provided ζ$_{K,n}$λ$_{K,n}$ n/ log n = O ( M 1+ δ n ). 

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Control of (65a): By Assumption 2(ii), the predictable variation of the summands in (65a) may be 

