(ii) We will prove the Theorem for the case analyzed in part (i)c; in the other cases the proof
follows straightforwardly and is therefore omitted. Under H a , we have that
d Xt = μ ( Xt )d t + yptW Wɪ ,t
σt = g( ft )
d ft = b ( ft )d t + σ ɪ ( ft )d W2 ,t.
Pointwise in r, we can rewrite Zn,m,r as
Zn,m,r
,— L ( n-ɪ) rj
m ∑ (Sn(Xi/n) - g (fi/n))
i =1
L ( m— ɪ) rj
- √m (x(j+ɪ)/m - Xj/m)
j=i
l (n- 1)rj
∑ g (fi/n)
i =1
ʌ- l(n-ɪ)rj
m ∑ (Sn(Xi/n) - g (fi/n))
i = 1
'------------------------------->z-------------------------------'
E ʌ-.
n,m,r
([ ( m— ɪ) rj
Σ (X(j+ɪ)/m - Xj/m)
j=1
I g ( fs )d s
Jo
4.
^>Z
Fm
√m L(n- ɪ)rj r
+— ∑ g (fi/n) - Vm g(fs)ds.
n i=1 Jo
'----------------------------------------4/-----------------------------------------'
(24)
Ln,m,r
By the same argument used in the proof of part (i)a, Step 4 and Step 2 (respectively) Ln,m,r =
op (1) and Fm,r = Op (1).
We can expand En,m,r as
E ™ l— L ( n-ɪ)rj n ⅛ i =1 |
^n = 11 1 {∖Xj∕n-Xi∕n∖<ξn}n (X(j + 1)/n Xj/n) _ ( f ) I n— ɪ d g ∖i⅛∣El ∖ j=1=1 1 {∖Xj∕n-Xi∕n∖<ξn} ) |
— l ( n-1)rj n ⅛ i =1 <- |
f∑n n-ɪ 1flv 2 2n ;2 n f( j+1) /n ( Xs X X1/n )√g^(7s)d Wɪ s ∖ j = 1 {∖Xj∕n-Xi∕n∖<ξn} j/n s j∕n sj ɪ,s I ^n—ɪ1 I ∖ 2~=i=i 1 {∖χj∕n-Xi∕n∖<ξn} __Z p n,m,r |
24