Download (535Kb) - LSE Research Online - London School of ...

Lemma 5.9 Let the random process (t) be de ned and continuous with probability 1 on the closed
set F . Assume that there exist integers m r 2 and a number H such that for all t s 2 F
Ej (t) ; (s)j m hjt ; sj r Ej (t)j m h:
Then
Efsup j (t)j
t2F
m g B0h (5.31)
where B0 depends on m r and does not depend on .
6 Second order expansions.
Since by Lemma 5.6 Um = ;Bm + Vm + 1+
m m, the expansion for Um requires one for Vm (4.8).
This in turn requires one for Z =(Z1Z2) 0 , where Zi are given in (4.6) and de ned with ` = 3 in case
of tapering and ` = 1 in case of no tapering. We assume in this section that (4.4) and Assumption
m are satis ed. We shall derive the expansion of Vm in terms of e m( m).
We shall approximate the distribution function P (Z x)x=(x1x2) 2 R2 ,by
where
F (x) =
Z
y x
(y : )K(y)dy (6.1)
(y : )=(2 ) ;1 j j ;1=2 exp(; 1
2 y0 ;1 y) y 2 R 2
is the density of a zero-mean bivariate Gaussian vector with covariance matrix
= e1+1 e1+2
e2+1 e2+2
= 1 ; 2
;2 9
where the elements of are de ned by (7.5) and related to Z1Z2 by
E[ZpZv] =ep+v +2(m=n) e(p + v ` )+o( m) (6.2)
(see Lemma 7.5). The polynomial K(y) isgiven by
where P (2) (y), P (3) (y) are polynomials de ned by
P (2) (y) =2
Hij(y) = (y : ) ;1 @ 2
K(y)=1+( m 1
)
n 2! P (2) ;1=2 1
(y)+m
3! P (3) (y) (6.3)
2X
ij=1
@yi@yj
e(i + j ` )Hij(y) P (3) (y) =2
2X
ijk=1
(y : ) Hijk(y) =; (y : ) ;1 @ 3
@yi@yj@yk
Theorem 6.1 Suppose that (4.4) and Assumptions f l m hold. Then
ei+j+kHijk(y)
(y : ) i j k =1 2:
sup jP (Z 2 B) ; F (B)j =
B
4
3 sup F ((@B)
B
2 )+o(e m) (6.4)
for any = m ;1; (0