PII. S0161171204301444 http://ijmms.hindawi.com © Hindawi Publishing Corp.
LARGE DEVIATIONS FOR EXCHANGEABLE OBSERVATIONS
WITH APPLICATIONS
JINWEN CHEN
Received 7 January 2003
We first prove some large deviation results for a mixture of i.i.d. random variables. Com-pared with most of the known results in the literature, our results are built on relaxing some restrictive conditions that may not be easy to be checked in certain typical cases. The main feature in our main results is that we require little knowledge of (continuity of) the compo-nent measures and/or of the compactness of the support of the mixing measure. Instead, we pose certain moment conditions, which may be more practical in applications. We then use the large deviation approach to study the problem of estimating the component and the mixing measures.
2000 Mathematics Subject Classification: 60F10, 60G09, 62G05, 62G20.
Our investigation on the LDP turns out to have some significant implications to the problem of making inference on the component and/or the mixing measures which are usually unknown or contain some unknown key parameters. Suppose we have a sample, (ξ1, . . . , ξn), of exchangeable observations. A typical way to generate such a sample is as follows: first choose a distributionPθat random from a family{Pθ, θ∈Θ} according to certain lawµ, whereΘis a space of parameters; then sample from this distribution. Thus it is easily seen that such a sample is typically suitable for making inference on the component distributionPθ. Suppose a realization of this sample is obtained, one can, by using standard statistical methods, make inference or estimate on the distributionPθfrom which this sample comes. However, from the point of view of Bayesian statistics, it is important to make inference or estimate on the mixing measure or the prior distribution in Bayesian terminology,µ. From such a sample given above, it is not hard to find unbiased, even uniformly minimum variance unbiased estimators of some functionals ofµthat play key roles in statistics. But, by the LLN for exchangeable random variables, such estimators are usually not consistent ones. Actually it is hard to find consistent estimators from a single sample. To construct a consistent estimator, a number of such samples are usually needed. From this point of view, inSection 3we propose estimators forµor its functionals and use a large deviation approach to show their consistency.
InSection 2we formulate the main large deviation results followed by some exam-ples.Section 3is devoted to the proof of these results. InSection 4we study the esti-mate problem.
2. Large deviations. We consider a mixture of general i.i.d. random fields and start with some notations. First, for a topological spaceY, denote byᏮY the Borelσ-field; M1(Y )is the space of all probability measures on(Y ,ᏮY). Now letXbe a locally convex
Hausdorff topological vector space with the conjugateXand letX0⊂Xbe a separable metric space in the relative topology. Let(Ω,Ᏺ)be a given measurable space,Θa Haus-dorff topological space, and{Pθ, θ∈Θ}a family of probability measures on (Ω,Ᏺ)
satisfying that for eachA∈Ᏺ, θ→Pθ(A)is measurable on(Θ,ᏮΘ). Let {ξt, t∈Zd}
(d≥1) be a family ofX0-valued random variables on(Ω,Ᏺ)which are i.i.d. under each
Pθ. Letµbe a Radon probability measure on(Θ,ᏮΘ)and define
P=
Pθµ(dθ). (2.1)
Finally, let{Λn, n≥1}be a sequence of finite subsets ofZdsatisfyingΛ
n↑Zd. Denote
by|Λn|the cardinality ofΛn. For eachn≥1, define
¯
ξn=Λ1
n
t∈Λn
ξt (2.2)
and letPθn=Pθ◦ξ¯−1
n ,Pn=P◦ξ¯n−1which are probability measures on (X0,ᏮX0). The aim of this section is to study the LDP of{Pn, n≥1}. The main technique we will use
(H) for everya >0,there is a compact set Ca⊂X0,such that
lim sup
n→∞
1
ΛnlogPnCc a
≤ −a. (2.3)
Our main result is the following.
Theorem2.1. LetPnbe defined as above and let (H) hold. If for eachλ∈X,
Mθ(λ)=
eλ,ξ0dPθ<∞, µ-a.s., (2.4)
then{Pn, n≥1}satisfies an LDP with a good rate function, that is, there is a function I:X0→[0,∞]such that for everyA∈ᏮX0,
− inf
x∈A0I(x)≤lim infn→∞ 1
ΛnlogPn(A)≤lim sup
n→∞
1
ΛnlogPn(A)≤ −inf
x∈A¯
I(x) (2.5)
andIhas compact level sets, that is, for anya >0,{x:I(x)≤a}is compact inX0, where
A0andA¯are the interior and closure ofA, respectively. In particular, if for allλ∈X,
M(λ)≡
eλ,ξ0dP <∞, (2.6)
then the above conclusions hold.
Remark2.2. Assumption (H) is not a restrictive one. Indeed, in our case, to obtain a full LD result with agood rate function, that is, withI having compact level sets, (H) is a necessary condition. IfX0is compact, (H) is trivial. Some practically used sufficient condition for (H) to hold can be found in [4,5,6]. In the case whereX=Rd, sup
θMθ(λ) <
∞for someλ >0 is sufficient for (H). In the case of a more general Banach space, see Example 2.6.
Remark2.3. As a special case, letX=M(E)be the space of all finite signed
mea-sures on some Polish spaceE, equipped with the weak topology,X0=M1(E), and let {ηt, t∈Zd}be a family ofE-valued random variables on(Ω,Ᏺ)that are i.i.d. under each Pθ,ξt=δηt is the Dirac measure onηt. Then under (H),{P ((1/|Λn|)
t∈Λnδηt ∈ ·), n≥ 1}satisfies an LDP, since in this case (2.4) is automatically satisfied. In particular, ifX0 is compact, then all the conditions inTheorem 2.1are satisfied.
The following are some typical examples that satisfy the conditions ofTheorem 2.1.
Example2.4. LetX=X0=Rand Pθ the Gaussian distribution with meanα and varianceσ2, andθ=(α, σ2)∈Θ⊂R×R
+, whereR+ is the space of strictly positive
real numbers. Then (2.4) is satisfied for anyλ∈R. In particular, ifΘis a bounded set, then (H) is also satisfied. IfPθis a Poisson distribution with meanθ∈Θ⊂R+, then we
Example2.5. LetPθ be the Bernoulli distribution with parameterθ∈(0,1). Then all the conditions inTheorem 2.1are satisfied.
In the cases wherePθis exponential or geometric, (2.4) is not satisfied. These cases will be treated in the next theorem.
Example2.6. The third application is to Banach-valued sequences. In this case,X0=
Bis assumed to be a separable Banach space with norm·and{ξn, n≥1}is anX 0-valued sequence.Sn=n
i=1ξi. Suppose (i) there is aµ-null setΘ0⊂Θsuch that the family{Pθ, θ∈Θc0}is tight, and (ii)eαξ1dP·L∞(µ)<∞, for allα >0. Note that de Acosta [4] proved that the family{P (n−1Sn∈ ·), n≥1}is exponentially tight. Thus ourTheorem 2.1and its proof imply that this family satisfies a full LDP with the good rate function given by
I(x)=sup
λ∈B
λ, x−L(λ) , (2.7)
whereL(λ)=logexpλ,ξ1dP·L∞(µ). This complements [4, Theorem 5.1] by providing the accompanying large deviation lower bounds, and thus the full LDP.
As pointed out in [7], some simple examples do not satisfy the moment condition (2.4) or (2.6). For example, letξ0 have exponential distribution with parameter θ >
θ0≥0 underPθ [7, Example 3.3]. Then it is easy to check that (2.4) does not hold. For such cases, it has been proved in [7] that ifΘis compact and {Pθn, n≥1, θ∈Θ}is exponentially continuous (to be stated below), then under (H){Pn, n≥1}also satisfies
an LDP. Based on this result, we can treat the case whereΘis not compact. To this end we first recall that the family{Pθn, n≥1, θ∈Θ}is said to beexponentially continuous if wheneverθn→θ, the family{Pθnn, n≥1}satisfies an LDP. We have the following.
Theorem2.7. Assume that (H) holds and that{Pθn, n≥1, θ∈Θ}is exponentially continuous. If for eachλ∈X, there existδ >0and a compact setΘδ⊂Θwithµ(Θδ) <1,
such that
e(1+δ)λ,ξ0dPθ<∞ forµ-almost allθ∈Θc
δ, (2.8)
then{Pn, n≥1}satisfies an LDP with a good rate function.
It is easy to check that (2.8) is satisfied in both the exponential and the geometric cases.
Using the same argument, whenΘis countable, we can prove the following.
Theorem2.8. LetΘbe countable, and for eachn≥1, letµnbe a probability measure on(Θ,ᏮΘ)satisfyinglimn→∞|Λn|−1logµn(θ)=0for eachθ. If the family{Pθn, n≥1, θ∈ Θ}satisfies that(1)for eachθ,{Pθn, n≥1}satisfies an LDP;(2)for anyλ∈X, there existδ >0and a finite setΘδ⊂Θsuch that (2.8) holds; and(3)the sequence defined by
Pn= θ∈Θ
Pn
θµn(θ), n≥1, (2.9)
For all the above results,Xcan be replaced by any of its dense linear subspaces. In the above theorems, the LD rate function should be
I(x)= sup
f∈Cb(X0)
f (x)−L(f ), (2.10)
where
L(f )=lim
n→∞
1
Λnlog
e|Λn|fdPn, (2.11)
which exists for f ∈Cb(X0); see Section 3. At the same time, under the conditions of any of these theorems, for eachθ∈Θ,{Pθn, n≥1}satisfies an LDP with the rate function given by
Iθ(x)= sup
f∈Cb(X0)
f (x)−Lθ(f ) , (2.12)
which has compact level sets, whereLθ(f )is defined similarly toL(f )withPnreplaced
byPθn. Our proofs of the theorems (Section 3) show that for someΘ0⊂Θwithµ(Θ0)=1,
L(f )=supθ∈Θ0Lθ(f ), which suggests that one may consider whether we have
I= inf
θ∈Θ0
Iθ. (2.13)
This turns out to be a minimax problem. Equation (2.13) need not be true in general. There are various conditions for a minimax theorem to hold; see [9] or [10]. Usually three types of conditions for Λ(θ, f )≡f (x)−Lθ(f ) for fixed x ∈X0 are involved: (i) compactness or connectivity of the domains of the two arguments of Λ; and (ii) convexity of Λ; (iii) certain continuity ofΛ. Simple examples show that only two of these conditions are insufficient for (2.13). From the results mentioned in [9,10], we can see that if one has compactness, then the solutions are satisfactory. The situation is very complicated in the noncompact cases. Here we will not give further discussion.
3. Proof of the large deviations. To use the asymptotic value method [1] for a mea-surable functionf on(X0,ᏮX0), define
Ln(f )=Λn−1log
e|Λn|fdPn, n≥1, (3.1)
and for a subsetΘ0⊂Θ, define
LΘ0,n(f )=Λn −1
log
Θ0 µ(dθ)
e|Λn|fdPn
θ, n≥1. (3.2)
The main task is to prove that for allf∈Cb(X0), the limit limn→∞Ln(f )exists. To this
end, we will use linear functionals to approximate bounded continuous functions. Set
Ꮽ=
min 1≤i≤m
λi,·+ci, λi∈X, ci∈R, m≥1
. (3.3)
Lemma3.1. LetΘ0be a measurable subset ofΘwithµ(Θ0) >0,λi∈X,1≤i≤m. If there existsδ >0such that (2.8) holds for everyλiandθ∈Θ0, then for anyci∈R, 1≤i≤m, andM >0, withg=min1≤i≤m(λi,·+ci), the limit
LΘ0(g∧M)=n→∞limLΘ0,n(g∧M) (3.4)
exists and is finite, wherea∧b=min(a, b).
To prove this lemma, we need the following result which was proved in [1] (see [1, Lemma L.6.1 and its proof]).
Lemma3.2. For fixedn≥1andθ∈Θ. Letf1, . . . , fnbe measurable and such that (1) for any linear combinationf=n
i=1αifi, the limit
Lθ(f )≡lim
n→∞
1
Λnlog
e|Λn|fdPθ (3.5)
exists;
(2) (d/dt)Lθ(tf+(1−t)g)|t=0 exists for each pair of convex combinations f =
n
i=1αifiandg=
n i=1βifi.
ThenLθ(min1≤i≤nfi)exists and there is a convex combinationf0=
n
i=1α0ifisuch that
Lθ
min 1≤i≤nfi
=inf
Lθ
n
i=1
αifi
, αi≥0, n
i=1
αi=1
=Lθf0
. (3.6)
Proof ofLemma3.1. Writegi= λi,·+ci. Then under our assumptions, for each
θ∈Θ0, the limits
Lθ
m
i=1
αigi+αm+1M
=lim
n→∞
1
Λnlog
e|Λn|(mi=1αigi+αm+1M)dPn θ
=
m
i=1
αici+αm+1M+log
emi=1αiλi,ξ0dPθ
(3.7)
are finite in some open convex setA1containing
A=
α1, . . . , αm+1
∈Rm+1:αi≥0, m+1
i=1
αi=1
(3.8)
and for any(α1, . . . , αm+1)and(γ1, . . . , γm+1)inA, if we setf=mi=1αigi+αm+1Mand
h=m
i=1γigi+αm+1M, then(d/dt)Lθ((1−t)f+th)|t=0exists. Hence fromLemma 3.2 we know that there existsgθ0=
m
i=1αigi+αm+1M with(α1, . . . , αm+1)∈A, such that
gθ
0∧g∧M=g∧Mand
h(θ)≡Lθ(g∧M)=lim
n→∞
1
Λnlog
e|Λn|(g∧M)dPn
θ =Lθ
g0θ
. (3.9)
continuous onΘk, and we can chooseΘkto be increasing. LetΘ∞=limk→∞Θk. Then µ(Θ∞)=µ(Θ0). DefineΘk,µ=Supp(µ|Θk), whereµ|Θkis the restriction ofµtoΘk. Then
Θk,µ↑andµ(Θk,µ)=µ(Θk). We will show that
LΘ0(g∧M)=lim
k→∞θsup∈Θk,µ
Lθ(g∧M)= sup
θ∈∪∞ k=1Θk,µ
h(θ). (3.10)
First notice thatg∧M=gθ
0∧g∧M≤gθ0for eachθ∈Θ0, thus, by (2.4),
lim sup
n→∞ LΘ0,n(g∧M)≤lim supn→∞
1
Λnlog
Θ0 µ(dθ)
e|Λn|g0θdPn θ
=lim sup
n→∞
1
Λnlog
Θ0 µ(dθ)
e|Λn|Lθ(g0θ)dPn θ
=lim sup
n→∞
1
Λnlog
Θ0
e|Λn|h(θ)µ(dθ)
=lim sup
n→∞ k→∞lim
1
ΛnΘ
k
e|Λn|h(θ)µ(dθ)≤lim
k→∞θsup∈Θk,µ
h(θ).
(3.11)
Notice that under (H) the sequence{Θ0P n
θµ(dθ), n≥1}is also exponentially tight; by
the above inequality and [1, Lemma 5.1] we know thatl=supθ∈∪∞
k=1Θk,µh(θ)is finite. Therefore for anyε >0, we can chooseθ0∈ ∪∞k=1Θk,µ such that
hθ0
=Lθ0(g∧M) > l−2ε. (3.12)
Supposeθ0∈Θk,µ. Then by the continuity ofhonΘk, we can choose a neighborhood Vθ0ofθ0, such that for anyθ∈Vθ0Θk,
h(θ) > hθ0
−2ε> l−ε. (3.13)
Thus by Jensen’s inequality, [1, Lemma 5.1], and Fatou’s lemma, we see that for suffi-ciently largeN,
lim inf
n→∞ LΘ0,n(g∧M)=lim infn→∞ LΘ0,n
(g∧M)∨(−N)
≥lim inf
n→∞
1
Λnlog
Vθ0Θk
µ(dθ)
e|Λn|[(g∧M)∨(−N)]dPn
θ
≥lim inf
n→∞
1
µVθ0Θk
1
ΛnV
θ0Θk
log
e|Λn|[(g∧M)∨(−N)]dPn θ
µ(dθ)
≥ 1
µVθ0Θk
Vθ0Θk
lim inf
n→∞
1
Λnlog
e|Λn|(g∧M)dPn θ
µ(dθ)
= 1
µVθ0Θk
Vθ 0Θk
h(θ)µ(dθ) > l−ε.
(3.14)
Proof ofTheorem2.1. Letg=min1≤i≤m(λi,·+ci)∈ᏭandM >0. Then for any
δ >0, by assumption, there isΘ0⊂Θwithµ(Θ0)=1 such that (2.8) holds for allθ∈Θ0. Thus, byLemma 3.1, the limitL(g∧M)=limn→∞Ln(g∧M)=LΘ0(g∧M)exists and is finite. Therefore from the proof of [1, Theorem T.1.3], we see that for allf∈Cb(X0), the limit
L(f )=lim
n→∞Ln(f ) (3.15)
exists and is finite. Hence by [1, Theorem T.1.2],{Pn, n≥1}satisfies an LDP with the
good rate function given by
I(x)= sup
f∈Cb(X0)
f (x)−L(f ). (3.16)
Proof ofTheorem2.7. From the proof ofTheorem 2.1, we know that it suffices to show that for allg∈Ꮽ andM >0, the limit L(g∧M)exists and is finite. To do this, letgi = λi,· +ci and g=g1∧ ··· ∧gm. Then under the assumptions of the theorem, there existδ >0 and a compact setΘδ⊂Θwith µ(Θδ) <1, and aΘ0⊂Θcδ
withµ(Θ0)=µ(Θδc)such that (2.8) holds for allθ∈Θ0. Then the lemma implies that
the limitLΘc
δ(g∧M)=ŁΘ0(g∧M) exists and is finite. By the exponential continuity of {Pθn, n≥1, θ∈Θ}and [4, Theorems 2.1 and 2.2], we know that the sequence
{ΘδP
n
θµ(dθ), n≥1}satisfies an LDP. Hence for anyf∈Cb(X0), the limit
LΘδ(f )=n→∞lim 1
Λnlog
Θδ
µ(dθ)
e|Λn|fdPn
θ (3.17)
exists and is finite. ThereforeLΘδ(g∧M)∨(−N)exists and is finite for anyN >0. Notice that{ΘδP
n
θµ(dθ), n≥1}is also exponentially tight; again by [1, Lemma 5.1] we know
that for sufficiently largeN,
lim
n→∞
1
Λnlog
Θδ
µ(dθ)
e|Λn|((g∧M)∨(−N))dPn θ−log
Θδ
µ(dθ)
e|Λn|(g∧M)dPn θ
=0.
(3.18)
This means that the limitLΘδ(g∧M)exists and is finite. Thus
L(g∧M)=lim
n→∞
1
Λnlog
Θδ
µ(dθ)
e|Λn|(g∧M)dPn
θ+
Θ0 µ(dθ)
e|Λn|(g∧M)dPn
θ
=LΘδ(g∧M)∨LΘc δ(g∧M)
(3.19)
exists and is finite.
Finally we proveTheorem 2.8. Letgi= λi,·+ci, 1≤i≤m,M >0,g=g1∧···∧gm. Then by our assumptions and the proof of the lemma, there is a finite setΘδ⊂Θ, such
that for anyθ∈Θc
δ, the limit
Lθ(g∧M)=lim
n→∞
1
Λnlog
e|Λn|(g∧M)dPn
exists and is finite, and there is agθ
0 =
m
i=1αigi+αm+1M withαi≥0,m+i=11αi=1, such thatLθ(g∧M)=Lθ(gθ0),g0θ∧g∧M=g∧M. Thus for eachθ∈Θcδ,
lim inf
n→∞ LΘcδ,n≥lim infn→∞
1
Λnlogµn(θ)
e|Λn|(g∧M)dPn
θ =Lθ(g∧M); (3.21)
that is,
lim inf
n→∞ LΘcδ,n(g∧M)≥sup
θ∈Θc δ
Lθ(g∧M). (3.22)
On the other hand,
lim sup
n→∞ LΘ
c
δ,n(g∧M)=lim supn→∞ 1
Λnlog
θ∈Θc δ
µn(θ)
e|Λn|(gθ0∧g∧M)dPn θ
≤lim sup
n→∞
1
Λnlog
θ∈Θc δ
µn(θ)
e|Λn|gθ0dPθn
=lim sup
n→∞
1
Λnlog
θ∈Θc δ
µn(θ)e|Λn|Lθ(g∧M)≤sup
θ∈Θc δ
Lθ(g∧M).
(3.23)
Hence the limit
LΘc
δ(g∧M)=n→∞limLΘcδ,n(g∧M)=sup
θ∈Θc δ
Lθ(g∧M) (3.24)
exists and is finite. Forθ∈Θδ, since{Pθn, n≥1}is exponentially tight and satisfies an
LDP, by [1, lemma 5.1], for largeN,
Lθ(g∧M)=lim
n→∞
1
Λnlog
e|Λn|(g∧M)dPn
θ (3.25)
exists and is equal toLθ((g∧M)∨(−N)), hence finite. Therefore, the limit
LΘδ(g∧M)=nlim→∞ 1
Λn
θ∈Θδ
µn(θ)
e|Λn|(g∧M)dPn
θ =max θ∈Θδ
Lθ(g∧M) (3.26)
exists and is finite and, furthermore,
L(g∧M)=LΘδ(g∧M)
∨LΘc δ(g∧M)
(3.27)
exists and is finite, completing the proof.
4. Estimate of the mixing measure. Now we turn to the study of estimating the mixing measure µ or some of its functionals such as expectation and moments. We adopt almost all the setting given inSection 2, except that we replace the set of indices
simply to make the notations simple. Letµθ=Pθ◦ξ−1
1 be the law ofξ1onX0under
Pθ. LetT be the mapping fromΘtoM1(X0)withT (θ)=µθ, whereM1(X0)is the space of probability measures on X0. DenoteT (µ)=µ◦T−1. We consider the problem of estimatingT (µ), instead ofµ, since we can, if needed, make a certain change of the space of parameters so that the mixing measure isT (µ). Let{ξi,j, 1≤j≤n}, 1≤i≤
n, ben independent copies of{ξi, 1≤i≤n}. We still denote byP the underlying probability law. Define, forn≥1,
Ln,i=n1
n
j=1
δξi,j, 1≤i≤n, Rn=n1
n
i=1
δLn,i. (4.1)
The following result implies thatRnis a (strongly) consistent estimator ofT (µ).
Theorem4.1. Assume (H). Thenlimn→∞Rn=T (µ)a.s. in the sense of weak
conver-gence.
Remark4.2. From this theorem it is easily seen that ifφ∈Cb(M1(X0)), then
lim
n→∞
1
n n
i=1
φLn,i=
φdT (µ)=
φµθµ(dθ) a.s. (4.2)
In particular, ifφ∈Cb(X0), then
lim
n→∞
1
n2
n
i,j=1
φξi,j=
µθ(φ)µ(dθ) a.s. (4.3)
This suggests consistent estimators for expectation or more general moments ofµ.
Obviously, the above theorem is a consequence of the following large deviation result.
Theorem4.3. Assume (H). Then{Qn=P◦Rn−1, n≥1}satisfies a full LDP onM1(X0) endowed with the weak topology, with the good rate functionJgiven byJ(R)=H(R, T (µ)), the relative entropy ofRwith respect toT (µ).
The proof ofTheorem 4.3is completed with the following three lemmas.
Lemma4.4. LetLn=(1/n)ni=1δξ
i. Thenlimn→∞P◦L− 1
n =T (µ)weakly.
Proof. For eachθ∈Θ, by applying Sanov’s theorem, it is not hard to prove that limn→∞Pθ◦L−n1=δµθ weakly. Thus for anyf∈Cb(M1(X0)),
lim
n→∞
f dP◦L−1
n =n→∞lim
µ(dθ)
f dPθ◦L−1
n =
fµθµ(dθ)=
f dT (µ), (4.4)
Lemma4.5. For anyf∈Cb(M1(X0)), the limit
Λ(f )≡lim
n→∞
1
nlog
enf (ν)Qn(dν) (4.5)
exists and equalslogefdT (µ). Moreover, for any pair f and g inCb(M
1(X0)), t→
Λ(tf+(1−t)g)is differentiable.
Proof. The first assertion is a consequence ofLemma 4.4, and thus the second assertion easily follows.
Lemma4.6. Assume (H). Then{Qn, n≥1}is exponentially tight onM1(X0).
Proof. For anya >0, from (H) we know that there is a compact subsetCaofM1(X0)
such that
PLn∈Cac
≤e−an ∀n≥1. (4.6)
Now define
Ka=
R∈M1
M1
X0
:RCakc ≤ak,∀k≥1
. (4.7)
ThenKa is compact inM1(M1(X0))and, for each 1≤i≤n,
PδLn,i∈Kac
=PδLn∈K
c a
=P
∃k≥1|δLn
Cakc ≥ak
=P∃k≥1|Ln∈Cakc
≤1−e−ane−an
(4.8)
by (4.6). Note thatKais convex. We have
PRn∈Kac
≤
n
i=1
PδLn,i∈KaC
≤1ne−e−−anan, (4.9)
which implies the exponential tightness of{Qn, n≥1}.
Now the proof ofTheorem 4.3is completed by applying [1, Theorem T.1.4], with the rate function being given by
J(R)= sup
f∈Cb(M1(X0))
f dR−Λ(f )
= sup
f∈Cb(M1(X0))
f dR−log
efdT (µ)
(4.10)
which is justH(R, T (µ)).
Proof ofTheorem4.1. Letd(·,·)be any metric onM1(M1(X0))that generates the weak topology. Then fromTheorem 4.1we know that for any >0, there is a constant
α>0 such that
PdRn, T (µ)≥≤e−αn ∀n≥1. (4.11)
Acknowledgment. This research was supported by the National Natural Science Foundation of China (NSFC 10371063).
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