provides that (# S $_{n}$) max x$_{n}$ ∈S$_{n}$ P ( {∣ ∣ ∣$^{1}$ n n ∑ i =1 G$_{i,n}$ ( x$_{n}$ ) ϵ$_{1}$$_{,i,n}$ ∣ ≥ Cζ$_{K,n}$λ$_{K,n}$ √ (log n ) /n } ∩ A n ∣ X n 1 ) ≲ n ν$_{1}$ + η$_{2}$$^{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 } (56) 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$_{n}$ } (57) for finite positive constants C$_{1}$, . . . , C$_{4}$ (independent of X$_{1}$, . . . , X$_{n}$ ). Thus (51a) vanishes asymptoti- cally for all sufficiently large C provided M$_{n}$ = O ( ζ − 1 $_{K,n}$λ − 1 K,n √ n/ (log n )). 2 2 Control of the leading term in (51b): First note that | G$_{i,n}$ | ≤ 2 ζ $_{K,n}$λ K,n by the Cauchy-Schwarz inequality and Assumption 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 ). Setting M 1+ δ n ≍ ζ$_{K,n}$λ$_{K,n}$ √ n/ log n trivially satisfies the condition ζ$_{K,n}$λ$_{K,n}$ √ n/ log n = O ( M 1+ δ n ). The condition M$_{n}$ = O ( ζ − 1 $_{K,n}$λ − 1 K,n √ n/ (log n )) is satisfied for this choice of M$_{n}$ provided ζ 2 $_{K,n}$λ 2 K,n ≲ ( n/ log n ) δ/ (2+ δ ) (cf. condition (i)). Finally, it is straightforward to verify that M$_{n}$ → ∞ as a consequence of condition (i). Thus, both (51a) and (51b) vanish asymptotically. This completes the proof in the i.i.d. case. P ( max x$_{n}$ ∈S$_{n}$ ∣ ∣$^{1}$ n n ∑ i =1 G$_{i,n}$ϵ$_{2}$$_{,i,n}$ ∣ ≥ Cζ$_{K,n}$λ$_{K,n}$ √ (log n ) /n ) ≲ ζ 2 $_{K,n}$λ 2 $_{K,n}$E [ | ϵ$_{i}$ |{| ϵ$_{i}$ | > M$_{n}$ } ] ζ$_{K,n}$λ$_{K,n}$ √ (log n ) /n ≤ ζ$_{K,n}$$_{λ$_{K,n}$}$√$_{n}$ √$_{log}$$_{n}$ E [ | ϵ$_{i}$ | 2+ $^{δ}${ | ϵ$_{i}$ | > M$_{n}$ } ] M 1+ δ n With weakly dependent data we use $^{P}$( A ∩ B ) ≤ $^{P}$( A ) to bound remaining term on the right- hand side of (47) by P ({ max x$_{n}$ ∈S$_{n}$ | q h ( x$_{n}$ ) | ≥ 2 Cζ$_{K,n}$λ$_{K,n}$ √ (log n ) /n }) ≤ P ( max x$_{n}$ ∈S$_{n}$ | ˜ b K $_{w}$( x$_{n}$ ) $^{′}${ ( ˜ B ′ w ˜ B$_{w}$/n ) − − I$_{K}$ } ˜ B ′ $_{w}$e/n | ≥ Cζ$_{K,n}$λ$_{K,n}$ √ (log n ) /n ) (58) K ′ ′ + P ( max x$_{n}$ ∈S$_{n}$ | ˜ b K $_{w}$( x$_{n}$ ) ′ B ′ $_{w}$e/n | ≥ Cζ$_{K,n}$λ$_{K,n}$ √ (log n ) /n ) . (59) It is now shown that a sufficiently large C can be chosen to control terms (58) and (59). 35