content_image/324772.jpg content_image/324773.jpg content_image/324777.jpg content_image/324779.jpg content_image/324780.jpg
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 )). Control of the leading term in (51b): First note that G$_{i,n}$ 2 ζ 2 $_{K,n}$λ 2 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. 

<img src='content_image/324772.jpg'>

<img src='content_image/324773.jpg'>

<img src='content_image/324777.jpg'>

With weakly dependent data we use $^{P}$( A ∩ B ) ≤ $^{P}$( A ) to bound remaining term on the right- hand side of (47) by 

<img src='content_image/324779.jpg'>

<img src='content_image/324780.jpg'>

It is now shown that a sufficiently large C can be chosen to control terms (58) and (59). 

