for some finite constant M. But Assumption 4(i) and (iii) imply that the right-
hand side of the last inequality in (45) is finite. This completes the proof. ¥
7 Appendix B: Proofs of Main Results
Proof of Theorem 1
For the desired result, it is enough to prove that
{ω ∈ Ω∣αz (G, H) (ω) ≤ x} ∈ Z,
(46)
for all x ∈ R. Since the partition Π = {∏ι,..., ∏i, ...}of Ω generates the sigma
field Z , we have
sup |(Pz (G ∩ H))(ω) - (PzG)(ω)(PzH)(ω)∣
G∈G, H∈H
=
i∈I
sup
G∈G, H∈H
P (G ∩ H ∩ Πi) P (G ∩ Πi) P (H ∩ Πi)
P∏ P∏ P∏
1Πi (ω),
where 1Πi (ω) denotes the indicator function that equals one if ω ∈ Πi, and
otherwise equals zero. Let I = {1, 2, ..., i, ...} be the set of positive integers, and
let
Ix = i ∈ I |
sup
G∈G, H∈H
P (G ∩ H ∩ Πi) P (G ∩ Πi) P (H ∩ Πi)
P∏ P∏ P∏
≤x .
Then, we have (46) because
{ω ∈ Ω∣αz (G,H) (ω) ≤ x} = ⅛∏ ∈ Z. ¥
Before we start proving the lemmas and theorems in Section 4, we introduce
some additional lemmas that are used repeatedly below. Recall that wit =
x01,it,x02,it , x03,it ,zi0 . We also repeatedly use the diagonal matrix DT defined
in Section 4.
Lemma 16 Suppose that Assumptions 1-8 hold. Define Ξ = Ξ1 + Ξ2, where
Ξ1
Ξ2
Γ22,22 |
Γ22,31 Γ22,32 | |||||
diag |
0k1 , 0 |
0 k21 , Γ22,31 Γ0 Γ22,32 |
Γ31,31 Γ31,32 Γ031,32 Γ32,32 |
,0k33,0kz ; | ||
/ Γθι |
,Θ1 |
ΓΘ1,Θ21 0 |
0 |
rΘ1,μ32 |
rΘ1,μ33 |
0 |
Γ0 ΓΘ1 |
,Θ21 |
ΓΘ21,Θ21 0 |
0 |
r‰,μ32 |
γθ2∣,∕,33 |
0 |
0 |
00 |
0 |
0 |
0 |
0 | |
0 |
00 |
0 |
0 |
0 |
0 | |
Γ0 ΓΘ1 |
Γ0 0 |
0 |
Γ g32,g32 |
Γ g32 ,g33 |
Γ | |
,μ32 |
θ21,μ32 |
+Γ μ32,μ32 0 |
+Γ μ32,μ32 |
g32 ,z | ||
Γ0 |
Γ0 0 |
0 |
Γ0 g32,g33 |
Γ g33,g33 |
Γ | |
ΓΘ1 |
,μ33 |
θ21,μ33 |
+Γ0 μ32,μ32 |
+Γ μ33,μ33 |
g33,z | |
0 |
00 |
0 |
Γ0 g32,z |
Γ0 g33,z |
Γ |
/
37
More intriguing information
1. Surveying the welfare state: challenges, policy development and causes of resilience2. The name is absent
3. The name is absent
4. The demand for urban transport: An application of discrete choice model for Cadiz
5. On the Existence of the Moments of the Asymptotic Trace Statistic
6. Delayed Manifestation of T ransurethral Syndrome as a Complication of T ransurethral Prostatic Resection
7. From Communication to Presence: Cognition, Emotions and Culture towards the Ultimate Communicative Experience. Festschrift in honor of Luigi Anolli
8. The name is absent
9. The name is absent
10. Spectral density bandwith choice and prewightening in the estimation of heteroskadasticity and autocorrelation consistent covariance matrices in panel data models