= 2
1 {|M_a|<gn}CT4(u)Lχ(r, u)du
f∞∞ 1 {∣u-a∣<ξn}LX (1 ,u )d u
''
Lχ(r, a)da + θa.s.(1)
= 2
a.s. 2
1 {∣zξn∣<ξn}σ4( a + zξn ) LX ( r, a + zξn )d z
f∞∞ 1 {∣zξn∣<ξn}LX(1, a + zξn)dz
Lχ(r, a)da + oa,s. (1)
Proof of Step 4:
Z∞
-∞
Lχ(r, a)2
MM d a.
(20)
Cn,r
1 L (n_ 1) rJ
l σ 2( Xs )d s
0
√= £ σ2(Xi/n) --n
V n
i =1
1 l ( n_ 1)rJ l ( n_ 1)rJ ∕∙( i+1) /n
-= £ σ2(Xi/n) -√n £ I σ2(Xs)ds
nn i =1 i =1 7i/n
l ( n_ 1)rJ r( i+1) /n
n £
i=1 i/n
(σ 2( Xi/n )
- σ2(Xs)) ds
(21)
and, given the Lipschitz assumption on σ2(∙), the last line in (21) is oP(1) by the same
argument as the one used in Step 1.
Given Steps 1-4 above, it follows that the quadratic variation process of Zn,r is given by
2 [∞ σ4 (a) Lx(r, a)da + 2 [∞ σ4 (a) Lχ(^ a).2 da - 4 [∞ σ4 (a) Lχ(^ a).2 da
-∞ -∞ LX (1, a) -
∞ LX (1, a)
= 2 [∞ σ4 (a) Lχ(r, a)(LX(1,a) - Lχ(r, a)) da. (22)
∞ -QQ LX (1 ,a)
The statement in the theorem then follows.
(i)b Without loss of generality, suppose that r < r'. By noting that
1 l ( n_ 1)rJ l ( n_ 1)rJ
-= £ sn (Xi/n ) -√n £ (Xi+1 /n - Xi/n )2
nn i=1 i =1
1 [( n_ 1) r' ] [( n_ 1) r' ]
= — £ sn(Xi/n) -Vn £ (Xi+1 /n - Xi/n)2,
n i=1 i=1
with Sn(Xi/n) = 0 and (Xi+1 /,n - Xi/n)2 = 0 for i > L(n - 1)rJ, the result then follows by
the continuous mapping theorem.
22