Notice that since ∣∣Z⅞τ (E⅛^χ3jit — Ex3jit) ∣∣4 ≤ τ (r)4 G (¾)4 by Assumption
6(iv), we have sup, ∣∣Z⅞feT (A¾y.⅛fe,i - Ex3kti) ∣∣4 ≤ (ʃθɪ τ (r)4 dr) supiG(¾)4
for к = 1,2, and 3. Therefore, by Lemma 10, we have
E KD3kτ (Ej^ix3k,i — Ex3k1i} K
(uniformly in i)
< ∞, for к = 1, 2, 3,
where τ3ι (r) = 0. So,
supE ∖∖D3τ (Eτz.x3,i - Ex3ji) ∣∣4 < ∞. (67)
Similarly, it follows that
sup sup E ∣∣Z⅛τ (E∏sx3 - E⅛)∣∣4 < ∞∙ (68)
N,T l<i<N
In addition, notice that
sup sup ∣∣Z⅛τ (Ex31i - Ex3)Il
N,Tl<i<N
sup sup
N,T l<i<N
(D33τ {Ex33tι — Ex33)
D3∙2τ (Ex321i — Ex32) — H32μg32i
D33τ {Ex33yι — Ex33) — H33μg33.
(69)
+ sup sup
N 3<i<N
0
-^32 At332
-^33 At333
By Assumption 6(v), as T → ∞,
sup sup
N 3<i<N
(D33τ (Ex33g. — Ex33)
D3,2τ (¾2,i — Ex32) — H32μg32 .
D33τ (Ex33ji — Ex33) — H33μg33 .
sup sup
D33τ {Ex33,i — Ex33j) ''
D32τ (Ex32,i — Ex32j) — H32 (μg32 i — At3323J
D33τ (Ex33ιi — Ex33j) — H33 ∖μg33 i — μ933 j) J
sup
i,j
f D33τ (Ex33ji — Ex33j)
D32τ (Ex32,i — Ex32j) — H32 ∖μg32 i — At332i3
D33τ (Ex33,i — Ex33j) — H33 [μg33 i — μ933 j
46
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