for some constant M2 < ∞. By Assumptions 2 and 6(i), for t = [Tr] and finite
constant M2 , we have
sup sup E kD1Tx1,itk4
i,T 1≤t≤T
≤ sup sup M3 (∣∣D1TIl4E ∣∣(xι,it
i,T 1≤t≤T
→ sup sup ∣τ1 (r) Θ1,i ∣4 < ∞.
i r∈[0,1]
Ex1,it)k4+ kD1TEx1,itk4
Next, similarly, by Assumptions 3(ii) and (iii),
sup sup E ∣x2,it ∣4 ≤ sup sup M3 E∣x2,it
i,T 1≤t≤T i,T 1≤t≤T
< ∞,
Ex2,itk4+kEx2,itk4
and by Assumptions 4(ii) and (iii) and 6(iii) and (iv),
sup sup Ekx3,itk4 ≤
i,T 1≤t≤T
sup
i,T
sup M3 E °x3,it - EFzi x3,it°4 + ∣Ex3,it∣4
1≤t≤T i
Therefore,
sup sup
i,T 1≤t≤T
which yields (83). ¥
Part (c)
By Lemma 16(a). ¥
Part (d)
By Lemma 16(d). ¥
M2
∞.
E (suΡi,t °x3,it - EFzi x3,it°4)
+ supi,t E ° ° EFzi x3,it - Ex3,it°4
+ supi,t ∣Ex3,it ∣4
kGx,T (xit
t
Xi)k2
2<M,
Proof of Lemma 3
By Lemma 16(c). ¥
Proof of Lemma 4
Write
., Pi Dtwi≈i=N Pi D
Twei
(ui
Ef.Ui) + (N Pi DT⅛W,'DT) λ. (85)
Notice that by Assumption 10,
∣Ef ɪ P
Fw ° N
i DT wei (ui
EFwui)
2 σ2 1
) = NutrE\N Pi DtυjiWj0 DTJ ■
54