Let X be a topological space. Let I :X → [0,∞] have compact sublevel sets. Let
(µn)n∈N be a sequence of probability measures onB(X).
4.B.1 Lemma. Let (Fm)m∈N be a decreasing sequence of closed sets with F = T
m∈NFm. Then
sup
m∈N
infI(Fm) = infI(F). (4.132)
§4.C. Proof of Theorem 4.9.2
Chapter
4
Proof. Letc:= supm∈NinfI(Fm). Note thatc≤infI(F). If c=∞there is nothing to prove. Assume thatc <∞. LetK be the compact set [I≤c]. Then Fm∩K6=∅ for allm∈N, whenceF∩K=6 ∅ and thusinfI(F)≤c.
4.B.2. For Lemma 4.B.1 the condition that I has compact sublevel sets is not re- dundant. For example: LetI:N∪{0} →[0,∞]be given byI(0) = 1andI(x) = 0for
x∈N. Then forFm={0}∪{m, m+1, . . .}andF ={0}one hassupm∈NinfI(Fm) = 0 andinfI(F) = 1.
4.B.3 Lemma.
(a) U be a set of open subsets ofX. Suppose that for allG∈ U
lim inf
n→∞ 1
nlogµn(G)≥ −infI(G). (4.133)
ThenG=SU satisfies (4.133)as well.10
(b) Let F1, F2, . . . be closed. Suppose that for all F ∈ {Fm:m∈N}
lim sup
n→∞ 1
nlogµn(F)≤ −infI(F). (4.134) ThenF =Tm∈NFm satisfies (4.134)as well.
Proof. lim inf n→∞ 1 nlogµn [ U≥sup G∈U lim inf n→∞ 1 nlogµn(G)≥sup G∈U (−infI(G)), (4.135) lim sup n→∞ 1 nlogµn \ m∈N Fm ≤ inf m∈Nlim supn→∞ 1
nlogµn(Fm)≤minf∈N(−infI(Fm)). (4.136) Now apply Lemma 4.B.1.
As a consequence of Lemma 4.B.3 we obtain the following.
4.B.4 Theorem. Suppose thatG is a basis for the topology onX, such that (4.133) holds for all G∈ G and (4.134) holds for F = X \G. Suppose that every open G
can be written as countable union of elements inG. Then (µn)n∈N satisfies the large
deviation principle with rate functionI.
§4.C Proof of Theorem 4.9.2
Proof of Theorem 4.9.2. AsX andY are separable metric spaces, every open subset of X × Y is a countable union of elements of the formA×B where A⊂ X is open andB∈ H, where (withdY the metric onY)
H={B(y, δ) :δ >0} ∪ {B(z, δ) :z6=y,0< δ < dY(y, z)}. (4.137)
Chapter
4
We use Theorem 4.B.4 to prove the large deviation bounds. Note first that(X × Y)\ (A×B) = (X ×(Y \B))∪((X \A)× Y), that min{infI(X \A),infL(Y \B)} = inf(x,y)∈(X ×Y)\(A×B)I(x) +L(y)and that by (4.129)
lim sup
n→∞ 1
nlogµn((X × Y)\(A×B))
≤maxnlim sup
n→∞ 1
nlogµn(X ×(Y \B)),lim sup n→∞
1
nlogµn((X \A)× Y)
o
.
Using this and Theorem 4.B.4 it is sufficient to show that for all open sets A ⊂ X
andB⊂ Y lim sup n→∞ 1 nlogµn(X ×(Y \B))≤ −infL(Y \B), (4.138) lim sup n→∞ 1 nlogµn((X \A)× Y)≤ −infI(X \A) (4.139) lim inf n→∞ 1
nlogµn(A×B)≥ −infI(A)−infL(B). (4.140) LetA⊂ X be open andB∈ H.
•(4.138) follows from the fact thatµn(X ×(Y \B)) =νn(Y \B).
•(4.139) follows from the fact that by (4.116), (4.118) and (4.129) we have
lim sup
n→∞ 1
nlogµn((X \A)× Y)
= maxnlim sup
n→∞ 1 nlogµ
1
n(X \A),lim sup n→∞ 1 nlogµ 2 n(X \A) o = lim sup n→∞ 1 nlogµ 1 n(X \A)≤ −infI(X \A), (4.141)
•(4.140) follows by separating two cases (as either y∈B ory /∈B):
Ify /∈B, thenWn∩B=∅and soµn(A×B) =µ1n(A)νn(B)for largen, whence
lim inf
n→∞ 1
nlogµn(A×B) = lim infn→∞
1 nlogµ 1 n(A) +n1logνn(B) . (4.142)
Suppose thaty∈B, i.e.,Wn⊂B for large n. By (4.130) we obtain
lim inf
n→∞ 1
nlogµn(A×B)
= maxnlim inf
n→∞ 1 nlogµ 1 n(A) +1nlog Z Wn αn dνn , lim inf n→∞ 1 nlogµ 2 n(A) +1nlog Z Wn (1−αn) dνn , lim inf n→∞ 1 nlogµ 1 n(A) +1nlog Z Wc n 1B dνn o (4.143) Using thatlim infn→∞n1log(
R
Wc
n1B dνn)≤0
together with (4.116) and (4.117), we obtain lim inf n→∞ 1 nlogµn(A×B) = max n lim inf n→∞ 1 nlogµ 1
n(A),lim inf n→∞ 1 nlogµ 2 n(A) o ≥ −infI(A). (4.144)
§4.D. Miscellaneous: Additional remarks
Chapter
4
BecauseinfL(B)≥0, we conclude (4.140).
We leave it to the reader to check thatηnis the weakly continuous product regular conditional probability underµn with respect to π.
§4.D Miscellaneous: Additional remarks
In Section 4.D.1 we comment on the conditions of Theorem 4.6.3 and show that one cannot expect a straightforward reduction of these conditions to be feasible. In Section 4.D.2 we comment on the difference between large deviations of sequences and large deviations of nets of probability measures.
§4.D.1 Regarding the conditions of Theorem 4.6.3
4.D.1. We make an observation regarding condition (a2) of Theorem 4.6.3, which reads as follows: For allU ∈ G
lim inf
n→∞ lim supV∈Vn
1
nlogµn(U × Y|X ×V)≥ −infI(U). (4.145) This holds if and only if ∀U ∈ G, ∀ε >0∃N ∀n≥N ∀V0∈ Vn ∃V ∈ Vn, V ⊂V0:
1
nlogµn(U× Y|X ×V)≥ −infI(U)−ε. (4.146) Note that 1 nlogµn(U× Y|X ×V) = 1 nlogµn(U×V)− 1 nlogµn(X ×V), (4.147)
infI(U) = infJ(U× {y})−infJ(X × {y}). (4.148) By the large deviation principle of (µn)n∈N and Lemma 4.5.3 we have ∀U ∈ G, ∀δ >0, ∃N ∀n≥N ∀V0∈ Vn ∃V ∈ Vn, V ⊂V0:
1
nlogµn(X ×V)≤ n1logµn(X ×V)≤ −infJ(X × {y}) +δ. (4.149) With this the following would be sufficient to imply condition (a2) of Theorem 4.6.3:
∀U ∈ G, ∀ε >0, ∃N ∀n≥N ∀V0∈ Vn ∃V ∈ Vn, V ⊂V0: 1
nlogµn(U ×V)≥ −infJ(U× {y})−ε. (4.150) We mention that we cannot expect this condition to hold in general. Indeed, ifY is first countable, then we may assume thatVnis a countable set consisting of decreasing sets Vn
m with
T
m∈NVmn={yn}, and therefore the above condition regarding (4.150) implies that for allU ∈ G with infJ(U× {y})<∞ and for nlarge enough µn(U×
{yn}) = limm→∞µn(U×Vmn)>0.
Chapter
4
§4.D.2 Large deviations over different parameters
In large deviation theory there exist basically two formulations of the large deviation principle. One is in terms of sequences, as in the books by den Hollander [50] and by Rassoul-Agha and Seppäläinen [75], the other is in terms of nets indexed by(0,∞), as in the books by Dembo and Zeitouni [24] and by Deuschel and Stroock [26]. In Theorem 4.D.2 we address the relation between these formulations.
Suppose that D ⊂(0,∞)is such that 0∈D. LetX be a topological space. Let
(µε)ε∈D be probability measures onB(X).
4.D.2 Theorem. ForA∈ B(X) lim inf
ε↓0 εlogµε(A)≥ −infI(A
◦) (4.151)
if and only if for all sequences (εn)n∈Nin D withεn →0
lim inf
n→∞ εnlogµεn(A)≥ −infI(A
◦). (4.152)
Similarly
lim sup
ε↓0
εlogµε(A)≤ −infI(A) (4.153) if and only if for all sequences (εn)n∈Nin D withεn →0
lim sup
n→∞ εnlogµεn(A)≤ −infI(A).
(4.154) Proof. Elementary.
§4.D. Miscellaneous: Additional remarks
Chapter