1 I<’n-1>rj (∑. =:; 1 {⅛∕n-,Y,∕n∣<M" U/n (σ2(ɪ.) - σ2(Xi/n)) d«
∑n-1
j = 1 1 {∣ Xj∙∕n~X≈∕n∣<ξn}
~^∕""~
Dn,r
Now, given Lemma 1, Dn,r = oa.s. (1), provided that n1 /2+εξn → 0, as n → ∞. It is immediate
to see that Hn,r is of a smaller order of probability than Gn,r.
Let < Gn >r denote the discretized quadratic variation process of Fn,r. By a similar argument
as in Bandi and Phillips (2003, pp.271-272),
we have that
plim n→∞
< Gn >r
- 2 ∞ σ4 ( a )
-∞
L X ( r, a )2
Lχ (1, a)
Thus, by the same argument as in the proof of Theorem 3 in Bandi and Phillips (2003), the
statement in Step 1 follows.
Proof of Step 2: It follows from Theorem 1 in Barndorff-Nielsen and Shephard (2004a).
Proof of Step 3: The discretized covariation process < An,Bn >r,
< An, Bn >
L(n-1’rj [(n-1’rj
i =1 j =1
'1 { ∣ .χiin-X∕,∣ <<n} J/N /n ( ' - χ∕n) σ(χ. )dW 2 '
∑n=1 1 { ∣ χ3∕n-χi∕n ∣ <ξn}
L(n-1’rj L(n-1’rj
= 2 n ∑ ∑
i =1 j =1
1 { ∣ χj∕n-χi∕n ∣ <ξn}σ 4( Xj/n + 0a-s-(1))
Vn-1 m, , ,
2=1=1 1 {∣ Xj∕n-Xi∕n ∣ <ξn}
(19)
= 2

1 {∖Xu-Xa∖<ξn}σ 4( XU )d U
Jθ1 1 {∖Xu-Xa∖<ξn} d U
d a + 0a.s. (1)
[∞ J ∣,∞ 1 u a <ξ. σ4l∏lχ(∣: l∏<∖u
J-oo -J-OO J-∞ 1 {∖u-a∖<ξn}LX(1 ,u)du
Lx(r, a)da + oa.s. (1),
where the 2 (instead of 4) on right hand side of (19) comes from Lemma 5.3 in Jacod and
Protter (1998). Along the lines of Bandi and Phillips (2001, 2003), by the change of variable
u-a
ξn
= z,
< An, Bn >
21
More intriguing information
1. Benefits of travel time savings for freight transportation : beyond the costs2. AN ECONOMIC EVALUATION OF THE COLORADO RIVER BASIN SALINITY CONTROL PROGRAM
3. The name is absent
4. Testing Gribat´s Law Across Regions. Evidence from Spain.
5. The Trade Effects of MERCOSUR and The Andean Community on U.S. Cotton Exports to CBI countries
6. Beyond Networks? A brief response to ‘Which networks matter in education governance?’
7. Direct observations of the kinetics of migrating T-cells suggest active retention by endothelial cells with continual bidirectional migration
8. Multifunctionality of Agriculture: An Inquiry Into the Complementarity Between Landscape Preservation and Food Security
9. The Effects of Attendance on Academic Performance: Panel Data Evidence for Introductory Microeconomics
10. The name is absent