3.2 Central limit theorem for Π
n
ǫ
n
In this subsection we establish a central limit theorem for Π
n
ǫ
n
. Set S
n
ǫ
n
= a
n
Π
n
ǫ
n
− EΠ
n
ǫ
n
σ
n
, where a
n
= nr
d n
1 4
and σ
2 n
= Var a
n
Π
n
ǫ
n
− EΠ
n
ǫ
n
. We shall verify that as n
→ ∞ S
n
ǫ
n D
→ N 0, 1. 3.13
To show this we require the following special case of Theorem 1 of Shergin 1990.
Fact 3.1. Let X
i,n
: i
∈ Z
d
denote a triangular array of mean zero m-dependent random fields, and let
J
n
⊂ Z
d
be such that i Var
P
i
∈J
n
X
i,n
→ 1 as n → ∞, and ii For some 2
s 3, P
i ∈J
n
E |X
i,n
|
s
→ 0 as n → ∞. Then
X
i ∈J
n
X
i,n
D
→ N 0, 1. We use Shergin’s result as follows. Recall the definition of
ǫ
n
in 3.1 and also that VarΠ
n
ǫ
n
= Z
˜ E
n
Z
˜ E
n
C ∆
n
x, ∆
n
y dxd y,
with ˜
E
n
= E
n
∩ S
r
n
f
. Next, consider the regular grid given by
A
i
= x
i
1
, x
i
1
+1
] × . . . × x
i
d
, x
i
d
+1
], where
i =i
1
, . . . , i
d
, i
1
, . . . , i
d
∈ Z and x
i
= i r
n
for i ∈ Z. Define
R
i
= A
i
∩ ˜ E
n
. With
J
n
= {i ∈ Z
d
: A
i
∩ ˜ E
n
6= ; } we see that {R
i
: i
∈ J
n
} constitutes a partition of ˜ E
n
. Note that for each
i ∈ J
n
, λ R
i
≤ r
d n
. We claim that for all large n
Card J
n
≤ C p
ǫ
n
r
−d n
. 3.14
2629
To see this, we use the fact that, according to 4.3, there exists ¯ ρ 0 such that for all large n,
˜ E
n
⊂ V
∂ S
f
, ¯ ρpǫ
n
. Thus, since r
n
pǫ
n
→ 0 by 3.2, [
i ∈J
n
A
i
⊂ V
∂ S
f
, ¯ ρ + 2pǫ
n
and, consequently,
r
d n
Card J
n
≤ λ V
∂ S
f
, ¯ ρ + 2pǫ
n
≤ C
p ǫ
n
. Keeping in mind the fact that for any disjoint sets B
1
, . . . , B
k
in R
d
such that, for 1 ≤ i 6= j ≤ k,
inf ¦
kx − yk : x ∈ B
i
, y ∈ B
j
© r
n
, then
Z
B
i
∆
n
xdx, i = 1, . . . , k, are independent, we can easily infer that
X
i,n
= a
n
Z
R
i
∆
n
x − E∆
n
x dx
σ
n
, i
∈ J
n
, constitutes a 1-dependent random field on Z
d
. Recalling that a
n
= nr
d n
1 4
and σ
2 n
→ σ
2 f
as n → ∞ by 3.5 we get, for all i ∈ J
n
, X
i,n
≤ a
n
σ
n
λR
i
≤ Cnr
3d n
1 4
. Hence, by 3.14,
X
i ∈J
n
E |X
i,n
|
5 2
≤ C Card J
n
nr
3d n
5 8
≤ Cnr
3d 2
n 1
2
. Clearly this bound when combined with r.iii and d
≥ 2, gives as n → ∞, X
i ∈J
n
E |X
i,n
|
5 2
→ 0, which by the Shergin Fact 3.1 with s = 5
2 yields S
n
ǫ
n
= X
i ∈J
n
X
i,n
D
→ N 0, 1. Thus 3.13 holds.
2630
3.3 Central limit theorem for L
Kategori