By Lemma 15 and the definitions of Dit , D2τ, and D3τ, each term in (64) is
finite. Thus,
sup sup E ∣∣I1,i,NTIl4 < ∞.
N,T 1≤i≤N
Next, we consider the second term in the right-hand side of the inequality
(63). For some constant M3,
sup sup E ∣∣I2,i,NTIl4
N,T 1<i<N
/ supN,Tsupι≤i≤N ∣∣D1T (Exι,i - EX1)∣∣4 ʌ
+ supn,t sup1<i<N ∣∣EX21,i - EX21k4
M3 +supi,τ e °d3T (e√x3,i - Ex3,i) °
(65)
+ supn,t sup1<i<N E ∣∣D3τ (Efzx3 - E⅝)∣∣
+ supN,τ sup1≤i≤N ∣∣D3τ (Ex3,i - ES3)∣∣
∖ +supn sup1≤i≤N e kzi- zl∣4 /
For the required result, we need to show that all of the terms in the right-hand
side of the inequality (65) are bounded. Notice that for some finite constant
M4,
∣∣D1τ (ExM - Ex1)∣∣4
H1
(θ1∙i - N X θ1∙
(uniformly in i as T → ∞)
≤ 2 ∣∣H1∣∣ sup I∣θ1,i∏ < M4,
i
and
∣∣Ex21,i - Ex21∣4
θ21,i - (N X θ21,i
i
(uniformly in i as T → ∞)
≤ 2sup ∣∣θ21,i∏4 < M4.
i
So,
sup sup ∣∣D1τ (Ex1,i - Ex1)∣∣4 , sup sup ∣∣Ex21,i - Ex21∣4 < ∞. (66)
N,T 1≤i≤N , N,T 1≤i≤N ,
By Assumption 6(iii), the followings hold uniformly in i almost surely as T →
∞ :
°d31T (EFzi x31,i - Ex31,i) ° → θ,
°D32τ (Ef2.x31,i - Ex31jj)° → ∣H32g32,i (zi)∣∣4 ,
°D33τ (Efz.x33,i - Ex33,i) ° → ∣∣H33g33,j (zi)∣∣4.
45
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