in Yosida and Hewitt 1952. Since lim
n →∞
[0; 1=n = 1 6= 0 = {0}; is not
countably additive. To see that it is pure nitely additive, let be any countably additive measure satisfying 066. Then
[0; 1] = {0} + lim
n →∞
1 − 0; 1=n] = 0
which implies that ≡ 0.
Theorem 2.2 Yosida and Hewitt, 1952, Theorem 1:19. Let be any non-negative
pure nitely additive measure; and be any non-negative countably additive mea- sure on space
X; F. Then for any ¿ 0; there exists A in F such that A
c
= 0; A ¡ ; where A
c
is the complement of A .
Theorem 2.3. Any non-negative measure on space X; F can be uniquely written
as the sum of a non-negative; countably additive measure
c
and a non-negative; pure nitely additive measure
p
.
Proof. This is a combination of Theorems 1:23 and 1:24 in Yosida and Hewitt 1952.
3. Equilibrium LDPs
Let XE be the space of all nitely additive, non-negative, mass one measures on E equipped with the projective limit topology, i.e., the weakest topology such that for all
Borel subset B of E, B is continuous in . Under this topology, XE is Hausdor. The -algebra B of space XE is the smallest -algebra such that for all Borel subset
B of E; B is a measurable function of .
It was incorrectly stated in Dawson and Feng 1998 that XE can be identied with M
1
E equipped with the -topology for large deviation purposes. In this section we will rst clarify the issues associated with equilibrium LDPs on the space XE, and then
establish the LDPs for equilibrium measures on M
1
E under the weak topology. Recall that the -topology on M
1
E is the smallest topology such that h; fi=
R
E
fx d x is continuous in for any bounded measurable function f on E. This topology is clearly
stronger than the weak topology, and is the same as the subspace topology inherited from XE. We use M
1
E to denote space M
1
E equipped with the -topology.
Theorem 3.1. Every element in XE has the following unique decomposition:
=
ac
+
s
+
p
; 3.1
where
p
is a pure nitely additive measure;
ac
and
s
are both countably additive with
ac
. ;
s
⊥ .
Proof. The result follows from Theorem 2.3 and the Lebesgue decomposition theorem.
Remark. An element in XE is a probability measure if and only if
p
E = 0.
For any in XE satisfying
ac
E ¿ 0, let
ac
·|E=
ac
·=
ac
E. Then it is clear that
ac
·|E is in M
1
E. is said to be absolutely continuous with respect to , still
denoted by . , if
B = 0 implies B = 0. For any two probability measures ; ; H
| denotes the relative entropy of with respect to . Dene I =
[H
|
ac
·|E − log
ac
E] if .
;
ac
E ¿ 0; 6∈ M
1
E; H
| if .
; ∈ M
1
E; ∞
else: 3.2
Remark. .
implies that
s
= 0.
Theorem 3.2. The family
{
;
;
} satises a LDP on XE with good rate function I.
Proof. Let
P =
{{B
1
; : : : ; B
r
}: r¿1; B
1
; : : : ; B
r
is a partition of [0; 1] by Borel measurable sets
}: 3.3
Elements of P are denoted by ; —; and so on. We say — ≻ i — is ner than . Then
P partially ordered by
≻ is a partially ordered right-ltering set. For every — = B
1
; : : : ; B
r
∈ P, let X
—
= x = x
B
1
; : : : ; x
B
r
: x
B
i
¿ 0; i = 1; : : : ; r;
r
X
i=1
x
B
i
= 1 :
For any = C
1
; : : : ; C
l
; — = B
1
; : : : ; B
r
∈ P; — ≻ , dene
—
: X
—
→ X ;
x
B
1
; : : : ; x
B
r
→ X
B
k
⊂ C
1
x
B
k
; : : : ; X
B
k
⊂ C
l
x
B
k
: Then
{X
—
;
—
; ; — ∈ J; ≻} becomes a projective system, and the projective limit of
this system can be identied as XE. For any nite partition = B
1
; : : : ; B
r
of E, and any in XE, let I
=
X
r k=1
B
k
log B
k
B
k
if . ;
∞ else;
where we treat c=0 as innity for c ¿ 0. Let
˜ I = sup
I ;
3.4 where the supremum is taken over all nite partitions of E.
By the standard monotone class argument indicator function over an interval can be approximated by bounded continuous functions pointwise. Hence, the restriction of
the -algebra B on M
1
E coincides with the Borel -algebra generated by the weak topology, and
;
;
is well-dened on space XE; B. By using Theorem 3:3 of Dawson and Gartner 1987, one gets that
;
;
satisties a LDP with good rate function ˜
I . Hence to prove the theorem it suces to verify that
I = ˜ I . This is true if is not absolutely continuous with respect to
since both are innity. The case of in M
1
E follows from Lemma 2:3 in Dawson and Feng 1998. Now assume .
and 6∈ M
1
E. Then we have
ac
E ¡ 1;
p
6= 0. If
ac
E = 0, then by applying Theorem 2.2, both I and ˜ I are innity. Next we
assume that =
ac
E is in 0; 1. By denition we have I ¿I for any nite
partition of E. Thus I ¿ ˜ I . On the other hand, for any n¿1 choose a set A
n
such that
A
n
¡ 1=n
2
;
p
A
c n
= 0. This is possible because Theorem 2.2 and
p
is pure nitely additive. It is clear that
T
n i=1
A
i
¡ 1=n
2
;
p
T
n i=1
A
i c
= 0. Hence by taking intersection, the sequence
{A
n
} can be chosen to be decreasing. For any nite partition = B
1
; : : : ; B
r
we introduce a new nite partition — = B
1
∩ A
n
; B
1
∩ A
c n
; : : : ; B
r
∩ A
n
; B
r
∩ A
c n
. Note that for any p
i
; x
i
¿ 0; i = 1; 2 we have the inequality
p
1
+ p
2
log p
1
+ p
2
x
1
+ x
2
6 p
1
log p
1
x
1
+ p
2
log p
2
x
2
: This implies
I 6I
—
and I
—
=
r
X
k=1
B
k
∩ A
c n
log B
k
∩ A
c n
B
k
∩ A
c n
+
r
X
k=1
B
k
∩ A
n
log B
k
∩ A
n
B
k
∩ A
n
¿
r
X
k=1
B
k
∩ A
c n
log B
k
∩ A
c n
B
k
∩ A
c n
+ A
n
log A
n
A
n
=
r
X
k=1
B
k
∩ A
c n
log B
k
∩ A
c n
B
k
∩ A
c n
+ A
n
log A
n
A
n
∨ :
Letting n go to innity, we get I
—
¿ lim
n →∞
r
X
k=1
B
k
∩ A
c n
log B
k
∩ A
c n
B
k
∩ A
c n
=
r
X
k=1
lim
n →∞
B
k
∩ A
c n
log B
k
∩ A
c n
B
k
∩ A
c n
¿
r
X
k=1
B
k
∩ F log B
k
∩ F
ac
B
k
; where F =
S
n
A
c n
. Since F
c
= 0, we have ˜
I ¿I
—
¿
r
X
k=1
B
k
log B
k ac
B
k
; which implies that
˜ I ¿ sup
r
X
k=1
B
k
log B
k ac
B
k
= I :
Lemma 3.3. Assume that there is a sequence of decreasing intervals A
n
such that the length of A
n
converges to zero as n goes to innity; A
n
¿ 0 for all n; and T
n
A
n
= {x
} with x
= 0. Then I dened above is not a good rate function on M
1
E.
Remark. Clearly a large class of probability measures including Lebesgue measure
satisfy the condition in Lemma 3.3. But pure atomic measures with nite atoms do not satisfy the condition.
Proof. We will construct a counter example. Assume that I is a good rate function.
Then for any ¿ 0, the level set = { ∈ M
1
E: I 6 } is a -compact set.
For any ∈ 0; 1; n¿1, choose
n
d x = f
n
x d x with
f
n
x = A
n A
n
x + 1
− 1
− A
n A
c n
x; where
A
is the indicator function of set A, and A
c
denotes the complement of A. By denition, we have
I
n
= H |
n
= Z
E
log 1
f
n
d x =
Z
A
n
log A
n
d x + Z
A
c n
log A
c n
1 −
d x 6
log 1
1 −
; which implies that
n
∈ with =log1=1−. Since compactness implies the sequential compactness see Ganssler, 1971, Theorem 2:6, the sequence
n
converges in topology to a measure in and thus in weak topology to the same measure.
For any continuous function g on E, one has lim
n →∞
h
n
; g i = lim
n →∞
Z
A
n
gx A
n
d x + Z
A
c n
1 − gx
A
c n
d x = gx
+ Z
{x }
c
1 − gx
d x; which implies that
n
converges weakly to =
{x }
d x + 1 −
d x. This lead to the contradiction
H | = ∞6:
Lemma 3.4. Assume
satises the condition in Lemma 3:3. Then it is impossible to
establish a LDP for the Poisson–Dirichlet distribution with respect to on M
1
E with a good rate function
.
Proof. Being the projective limit of a system of Hausdor space, the space XE is
also Hausdor. By Tychono theorem, the product space
∈ P
X is compact. since
X E is a closed subset of
∈ P
X , it is also compact. Thus XE is a regular
topological space. Assume that a LDP is true on M
1
E with a good rate function J . By Lemma 3.3, J must dier from I . But the following arguments will lead to
the equality of the two which is an obvious contradiction. Fix an
in M
1
E. The LDP on M
1
E with good J implies the corresponding LDP on M
1
E with good J . Since M
1
E is regular, we get that for any ¿ 0 there is an open neighborhood G
w
of such that
inf
∈ G
w
J ¿J − ∧
1 ;
where G
w
is the closure in space M
1
E. Let G
X E
be an open set in XE such that G
X E
∩ M
1
E = G
w
. This is possible because the subspace topology on M
1
E inherited from XE is stronger than the weak topology. Now from the two LDPs, we get
− inf
G
w
I 6 − inf
G
X
I 6 lim inf
→0
log
;
;
G
X E
= lim inf
→0
log
;
;
G
w
6 lim sup
→0
log
;
;
G
w
6 − inf
G
w
J ; which implies that
I ¿ inf
G
w
I ¿ inf
G
w
J ¿J − ∧
1 :
Letting approach zero we end up with I ¿J
: On the other hand, since XE is also regular, we get that for any ¿ 0, there exists open set G
X E
containing such that
inf
∈ G
X E
I ¿I − ∧
1 and G
X E
is the closure in XE. Let G = G
X E
∩ M
1
E. Then as before we get − inf
G
J 6 lim inf
→0
log
;
;
G = lim inf
→0
log
;
;
G
X E
6 lim sup
→0
log
;
;
G
X E
6 − inf
G
X
E
I ; which implies that
J ¿ inf
G
J ¿ inf
G
X E
I ¿I − ∧
1 and J
¿I .
From Theorem 3.1 and Lemma 3.4 we can see that in order to get an equilibrium LDP in the topology, one has to expand M
1
E to a bigger space. Next we are going to show that under a weaker topology, the weak topology, the equilibrium LDP holds
on M
1
E. First note that the space M
1
E is a compact, Polish space with Prohorov metric . Hence the sequence
;
;
is exponentially tight. By Theorem 2.1, to obtain a LDP
for
;
;
with a good rate function it suces to verify that there exists a function J such that for every
∈ M
1
E, lim
→0
lim inf
→0
log
;
;
{; ¡ } = lim
→0
lim sup
→0
log
;
;
{; 6} = −J : 3.5
By Theorem 2.1, the function J is the good rate function. Let supp denote the support of a probability measure , and M
1;
E = { ∈
M
1
E: supp ⊂ supp
}. Let {
n
} be an arbitrary sequence in M
1;
E that con- verges to a in M
1
E. Since supp is a closed set, we get
1 = lim sup
n →∞
n
{supp }6{supp
}; which implies that
∈ M
1;
E. Hence M
1;
E is a closed subset of M
1
E. Next we prove 3.5 for
J = H
| if ∈ M
1;
E; ∞
else; We will treat 0 log
as zero. For any
∈ M
1
E, dene E =
{t ∈ 0; 1: {t}=0}. For any t
1
¡ t
2
¡ · · · ¡ t
k
∈ E ,
set
t
1
; :::; t
k
= [0; t
1
; : : : ; [t
k
; 1] which can be viewed as a probability measure on space
{0; 1; : : : ; k} with a probability [t
i
; t
i+1
at i 6= k and [t
k
; 1] at k. Set t = 0.
Lemma 3.5. For any ;
∈ M
1
E; H
| = sup
t
1
¡t
2
¡ ···¡t
k
∈ E ; k¿1
H
t
1
; :::; t
k
|
t
1
; :::; t
k
: 3.6
Proof. By Lemma 2:3 in Dawson and Feng 1998, we have
H |¿
sup
t
1
¡t
2
¡ ···¡t
k
∈ E ; k¿1
H
t
1
; :::; t
k
|
t
1
; :::; t
k
: 3.7
On the other hand, by 2.5 for any ¿ 0, there is a continuous function g on E such that
H |6
Z g d
− log Z
e
g
d + :
Now choose t
n 1
¡ t
n 2
· · · ¡ t
n k
n
in E such that
lim
n →∞
max
i=0;:::; k
n
−1
|t
n i+1
− t
n i
| + max
t; s ∈ [t
n i
; t
n i+1
]
|gt − gs| = 0:
This is possible because E is a dense subset of E. Choose n large enough and let
t
n
= 0, we get H
| 6
k
n
X
i=0
gt
n i
[t
n i
; t
n i+1
+ gt
n k
n
[t
n k
n
; 1]
−log
k
n
−1
X
i=0
e
gt
n i
[t
n i
; t
n i+1
+ e
gt
n kn
[t
n k
n
; 1] + +
n
g
6 sup
i
; i=0;:::; k
n
k
n
X
i=0 i
[t
n i
; t
n i+1
+
k
n
[t
n k
n
; 1]
−log
k
n
−1
X
i=0
e
i
[t
n i
; t
n i+1
+ e
kn
[t
n k
n
; 1] + +
n
g = H
t
n 1
;:::;t
kn
|
t
n 1
;:::;t
kn
+ +
n
g; where
n
g converges to zero as n goes to innity. Letting n go to innity, then go to zero, we get
H |6
sup
t
1
¡t
2
¡ ···¡t
k
∈ E ; k¿1
H
t
1
; :::; t
k
|
t
1
; :::; t
k
: This combined with 3.7 implies the result.
For any ¿ 0; ∈ M
1
E, let B; =
{ ∈ M
1
E: ; ¡ };
B; = { ∈ M
1
E: ; 6 }:
Since the weak topology on M
1
E is generated by the family { ∈ M
1
E: f ∈ C
b
E; x ∈ R; ¿ 0; |h; fi − x| ¡ };
there exist f
1
; : : : ; f
m
in C
b
E and ¿ 0 such that { ∈ M
1
E: |h; f
j
i − h; f
j
i| ¡ : j = 1; : : : ; m} ⊂ B; : Let
C = sup {|f
j
x |: x ∈ E; j = 1; : : : ; m};
and choose t
1
; : : : ; t
k
∈ E such that
sup {|f
j
x − f
j
y |: x; y ∈ [t
i
; t
i+1
]; i = 0; 1; : : : ; k; t
k+1
= 1; j = 1; : : : ; m } ¡ =4:
Choosing 0 ¡
1
¡ =2k + 1C, dene V
t
1
;:::;t
k
;
1
= { ∈ M
1
E: |[t
k
; 1] − [t
k
; 1] | ¡
1
; |[t
i
; t
i+1
− [t
i
; t
i+1
| ¡
1
; i = 0; : : : ; k − 1}:
Then for any in V
t
1
;:::; t
k
;
1
and any f
j
, we have |h; f
j
i − h; f
j
i| = Z
[t
k
;1]
f
j
xd x − dx
+
k −1
X
i=0
Z
[t
i
; t
i+1
f
j
xd x − dx
¡ 2
+
k
X
i=0
|f
j
t
i
|
1
¡ ;
which implies that V
t
1
;:::; t
k
;
1
⊂{ ∈ M
1
E: |h; f
j
i − h; f
j
i| ¡ : j = 1; : : : ; m} ⊂ B; : Let
F = [0; t
1
; : : : ; [t
k
; 1]: Then
;
;
◦ F
−1
is a Dirichlet distribution with parameters =
[0; t
1
; : : : ; [t
k
; 1]. By applying Theorem 2.2 in Dawson and Feng 1998 we get that for in M
1;
E, − J 6 −H
t
1
; :::; t
k
|
t
1
; :::; t
k
6 lim inf
→0
log
;
;
{V
t
1
;:::; t
k
;
1
} 6
lim inf
→0
log
;
;
{B; }: 3.8
Letting go to zero, we end up with − J 6 lim
→0
lim inf
→0
log
;
;
{B; }: 3.9
For other , 3.9 is trivially true. On the other hand, for any t
1
; : : : ; t
k
in E , we claim that the vector function F is
continuous at . This is because all boundary points have -measure zero. Hence for any
2
¿ 0, there exists ¿ 0 such that B;
⊂ V
t
1
;:::; t
k
;
2
: Let
V
t
1
;:::; t
k
;
2
= { ∈ M
1
E: |[t
k
; 1] − [t
k
; 1] |6
1
; |[t
i
; t
i+1
− [t
i
; t
i+1
|6
1
; i = 0; : : : ; k − 1}:
Then we have lim
→0
lim sup
→0
log
;
;
{ B; }6 lim
→0
lim sup
→0
log
;
;
{ V
t
1
;:::;t
k
;
2
}: 3.10 By letting
2
go to zero and applying Theorem 2.2 in Dawson and Feng 1998 to
;
;
◦ F
−1
again, one gets lim
→0
lim sup
→0
log
;
;
{ B; }6 − J
t1; :::; tk
t
1
; :::; t
k
; 3.11
where J
t1; :::; tk
t
1
; :::; t
k
= H
t
1
; :::; t
k
|
t
1
; :::; t
k
if
t
1
; :::; t
k
.
t
1
; :::; t
k
; ∞
else: Finally, taking supremum over the set E
and applying Lemma 3.5, one gets lim
→0
lim sup
→0
log
;
;
{ B; }6 − J ; 3.12
which, combined with 3.9, implies the following theorem.
Theorem 3.6. The family
{
;
;
} satises a LDP on M
1
E with good rate function J .
Remark. 1. By 3.2, for any in M
1
E, we have that I = ∞ if is not absolutely
continuous with respect to . On the other hand, if we choose =
1 2
+
1 2
, then is not absolutely continuous with respect to
but J ¡ ∞. Hence the restriction of
I on M
1
E is not equal to J . 2. Let
1
;
2
; : : : be a probability-valued random variable that has the Poisson– Dirichlet distribution with parameter =
cf. Kingman, 1975, and
1
;
2
; : : : ; be i.i.d. with common distribution
. Then
;
;
is the distribution of P
∞ i=1
i
i
cf. Ethier and Kurtz, 1994, Lemma 4:2, and the LDP we obtained describes the large deviations
in the following law of large numbers:
∞
X
i=1 i
i
⇒ :
The new features are clearly seen by comparing this with the Sanov theorem that describes the large deviations in the law of large numbers:
n
X
i=1
1 n
i
⇒ :
Corollary 3.1. The family
{
;
; ;V
} satises a LDP on space M
1
[0; 1] with good rate function J
V
= sup {V − J } − V − J .
Proof.
By Lemma 4.2 of Ethier and Kurtz 1994, one has
;
; ;V
d = Z
−1
exp V
;
;
d; 3.13
where Z is the normalizing constant. Since V x is continuous, we get that V
∈ CM
1
[0; 1]. By using Varadhan’s Lemma, we have
lim
→0
log Z = lim
→0
log Z
e
V = ;
;
d = sup {V − J }:
3.14 By direct calculation, we get that for any
∈ M
1
E lim
→0
lim inf
→0
log Z
B;
e
V = ;
;
d =V + lim
→0
lim inf
→0
log
;
;
{B; } =V + lim
→0
lim sup
→0
log
;
;
{ B; } = lim
→0
lim inf
→0
log Z
B;
e
V = ;
;
d =V
− J ;
which combined with 3.14 implies lim
→0
lim inf
→0
log
;
; ;V
{; ¡ } = lim
→0
lim sup
→0
log
;
; ;V
{; 6} = −J
V
: 3.15
Since the family {
;
; ;V
} is also exponentially tight, using Theorem 2.1 again, we get the result.
4. LDP for FV process with nite types
