Let Xh,2,it be the hth element of Xh,2,it. The required result follows if we can
show that
sup
i,T
√T X (xh,2,it
T t
- Exh,2,it)
< M for all h.
4
(41)
The proof of (41) is similar to the proof of Lemma 1 of Andrews (1991). This
proof relies on the following α- mixing inequality presented in Hall and Heyde
(1980, p. 278). Suppose that Y and W are random variables that are G-
measurable and H- measurable, respectively, with E ∣Y ∣p < ∞ and E ∣W∣q <
∞, where p,q > 1 with 1/p + 1/q < 1. Then,
∣E (Y - EY ) (W - EW ) ∣ ≤ 8 H Y ∣∣p ∣∣ W ∣∣g [α (G, H)]1-1/p-1/q , (42)
where α (G, H) is the α-mixing coefficient between the sigma fields G and H.
Now, let Xit = xh,2,it - Exh,2,it. Notice that
4
p e .. Σ X«)
sup T χ x x x∣e (Xi,XisXipXik )∣
i,τ t=1 s = 1 p=1 k= 1
T -tT-sT -p
⅛ 5⅛2χχχΣ∣E (XitXi,t+sXi,t+s+pXi,t+s+p+k) ∣
i,T t s=0 p=0 k=0
4!siuτp T X
Σ
0≤p,k≤s
0≤p+k+s≤T-t
∣ E (Xit (Xi,t+sXi,t+s+pXi,t+s+p+k)) ∣
+4!sup ɪ X X
i,T t 0≤s,k≤p
0≤p+k+s≤T-t
+4!sup .=2 X X
i,i t 0≤s,k≤p
0≤p+k+s≤T-t
1
+4!siup .χ
x
0≤s,p≤k
0≤p+k+s≤T-t
= I + II + III + IV, say.
E [(Xit
Xi,t+s) (Xi,t+s+pXi,t+s+p+k)]
-E (XitXi,t+s) E (Xi,t+s+pXi,t+s+p+k )
∣ E (XitXi,t+s) E (Xi,t+s+pXi,t+s+p+k) ∣
E ((Xit
Xi,t+sXi,t+s+p') Xi,t+s+p+k) ∣
By applying the inequality of (42) to Xi,t+sXi,t+s+pXi,t+s+p+k and Xit and
then by the Holder inequality, we have
1T
i ≤ 4!8sup .2 HXitk4q HXi,t+sH4q Il Xi,t+s+p ∣∣ 4q
i,T t=10≤p,k≤s≤T-t
×∣∣Xi,t+s+p+k∣∣4q αi (s)q-1
35
More intriguing information
1. National curriculum assessment: how to make it better2. Foreign direct investment in the Indian telecommunications sector
3. On the Integration of Digital Technologies into Mathematics Classrooms
4. Prevalence of exclusive breastfeeding and its determinants in first 6 months of life: A prospective study
5. The use of formal education in Denmark 1980-1992
6. Whatever happened to competition in space agency procurement? The case of NASA
7. Synthesis and biological activity of α-galactosyl ceramide KRN7000 and galactosyl (α1→2) galactosyl ceramide
8. The name is absent
9. Feeling Good about Giving: The Benefits (and Costs) of Self-Interested Charitable Behavior
10. Tastes, castes, and culture: The influence of society on preferences