final report - probability.ca

Proof.
V (Y 1 ) − V (x 1 ) = V ′ (x 1 ) (Y 1 − x 1 ) + 1 2 V ′′ (Z 1 ) (Y 1 − x 1 ) 2 ,
for some Z 1 ∈ (x 1 , Y 1 ) or (Y 1 , x 1 ). Thus,
where K is an upper bound for V ′′ .
dE Y1 [V (Y 1 ) − V (x 1 )] ≤ Kdl2
d − 1 ,
Lemma 1.12. (Lemma 2.7 in [RGG97]) Suppose V ∈ Cc
∞ is a function of the first component of Z d . Then
∣ ( ∣∣Gd
sup V x (d)) − GV (x 1 ) ∣ −−−→ d→∞
0.
x (d) ∈F d
Proof. After decomposing the proposal Y (d) into
(
Y 1 , Y (d)−) as
⎡
(
G d V x (d)) = dE Y1
⎣ ( ⎡ ⎛
(
V Y (d)) (
− V x (d))) E Y (d)−
⎣min ⎝1,
(
f(Y1)
f(x 1)
π
(Y (d)) ⎞
π ( x (d)) ⎠
one can focus on the inner ) expectation of this term. Let the inner expectation be denoted by E (Y 1 ), so denoting
with ε (Y 1 ) = log , we have
E (Y 1 ) = E
= E
[
[
min
min
(
(
1, exp
1, exp
for some Z i ∈ (x i , Y i ) or (Y i , x i ). By Proposition 1.8,
[ ( {
d∑
∣ E (Y 1) − E min 1, exp ε (Y 1 ) +
{
{
ε (Y 1 ) +
ε (Y 1 ) +
})]
d∑
(log f (Y i ) − log f (x i )) ,
i=2
d∑
(log f (x i )) ′ (Y i − x i )
i=2
+ 1 2 (log f (x i)) ′′ (Y i − x i ) 2 + 1 6 (log f (Z i)) ′′′ (Y i − x i ) 3 ]}]
,
i=2
[
(log f (x i )) ′ (Y i − x i ) −
≤ E [|W d |] + sup ∣ ′′′ ∣ (log f (z)) z∈R
where W d is as defined in Lemma 1.9. Furthermore, the term
∣ [ ( {
∣∣∣∣ d∑
sup E (Y 1 ) − E min 1, exp ε (Y 1 ) +
x (d) ∈F d
i=2
1 4l 3
,
6 (d − 1) 1 2
(2π) 1 2
[
(log f (x i )) ′ (Y i − x i ) −
⎤⎤
⎦⎦ ,
l 2 (
(log f (xi )) ′) ] })]∣ ∣
2 ∣∣∣
2 (d − 1)
Denote this term by ϕ (d), which converges to zero as d → ∞.
However,
d∑
[
ε (Y 1 ) + (log f (x i )) ′ l 2 (
(Y i − x i ) − (log f (xi )) ′) ]
2
,
2 (d − 1)
i=2
l 2 (
(log f (xi )) ′) ] })]∣ ∣
2 ∣∣∣
.
2 (d − 1)
is distributed according N ( )
ε (Y 1 ) − l 2 R d /2, l 2 R d , so that by Proposition 1.10,
[ ( {
d∑
[
∣ E − E min 1, exp ε (Y 1 ) + (log f (x i )) ′ l 2 (
(Y i − x i ) − (log f (xi )) ′) ] })]∣ ∣
2 ∣∣∣
2 (d − 1)
i=2
∣
≤ E [|W d |] + sup ∣(log f (z)) ′′∣ ∣
z∈R
12
1 4l 3
.
6 (d − 1) 1 2
(2π) 1 2