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Variational form of the large deviation functional
ARTICLE IN PRESS
Statistics & Probability Letters 77 (2007) 931–936 www.elsevier.com/locate/stapro
Variational form of the large deviation functional$ Henri Comman Department of Mathematics, University of Santiago de Chile, Bernardo O’Higgins 3363, Santiago, Chile Received 16 August 2005; received in revised form 27 January 2006; accepted 17 January 2007 Available online 9 February 2007
Abstract Under exponential tightness hypothesis, we show that the large deviation functional LðÞ has the same variational form as when large deviations hold. As an application, we identify the affine regularization of a rate function on any real Hausdorff topological vector space in terms of LðÞ. Some classical results involving the generalized log-moment generating function are improved. r 2007 Elsevier B.V. All rights reserved. Keywords: Large deviations; Generalized log-moment generating function
1. Introduction Let ðma Þ be a net of Borel probability measures on a topological space X, and let ðta Þ be a net in 0; þ1½ converging function h on X, we write mtaa ðeh=ta Þ for R hðxÞ=t to 0. taFor any ½1; þ1½-valued Borel tmeasurable h=ta a a ð X e ma ðdxÞÞ , and define LðhÞ ¼ log lim sup ma ðe Þ, and LðhÞ ¼ log lim mtaa ðeh=ta Þ when this limit exists. We denote by CðX Þ the set of ½1; þ1½-valued continuous functions on X. The usual version of Varadhan’s theorem for X Polish asserts that under exponential tightness hypothesis, a large deviation principle for ðmtaa Þ with rate function J implies the existence of LðhÞ for all h 2 CðX Þ satisfying some tail condition, with LðhÞ ¼ supfhðxÞ JðxÞg.
(1)
x2X
In this note, we show that if the large deviation hypothesis is dropped, then (1) remains true with LðhÞ in place of LðhÞ, and J replaced by the function l 0 , where l 0 ðxÞ ¼ log infflim sup mtaa ðGÞ : x 2 G X ; G openg for all x 2 X (note that a large deviation principle on a regular space is always governed by l 0 ); furthermore, this is valid for any topological space in which compact sets are Borel sets (Theorem 1). If moreover the exponential tightness is dropped, then the result remains true replacing CðX Þ by the set CK ðX Þ of elements h 2 CðX Þ for which fy 2 X : ehðxÞ pehðyÞ pehðxÞ þ g is compact for all x 2 X and 40 with ehðxÞ 4. A general version of Varadhan’s theorem is a direct consequence (Corollary 1). $
This work was supported in part by USACH-DICYT Grant 040533C. E-mail address: [email protected].
0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2007.01.005
ARTICLE IN PRESS H. Comman / Statistics & Probability Letters 77 (2007) 931–936
932
In the last section we apply Theorem 1 to the case where X is a real Hausdorff topological vector space and h belongs to its topological dual X . Assuming the exponential tightness and large deviations for ðmtaa Þ with rate function J, we show that the affine regularization of J is the Legendre–Fenchel transform L~ of the map L~ ~ defined on X by LðlÞ ¼ limM!þ1 Lðl1flpMg 11fl4Mg Þ. It follows that J ¼ L~ when J is convex and X locally convex. We show that some classical results (and in particular the equality J ¼ L or even J ¼ L ) known when Lo1 are still valid under weaker conditions (Corollaries 2, 3); in particular, Baldi’s theorem is slightly improved (Theorem 2). 1.1. Notations We recall that ðma Þ satisfies a large deviation principle with powers ðta Þ if there exists a ½0; þ1-valued lower semi-continuous function J on X such that lim sup mtaa ðF Þp sup eJðxÞ
for all closed F X
sup eJðxÞ p lim inf mtaa ðGÞ
for all open G X ;
(2)
x2F
and x2G
J is called a rate function for ðmtaa Þ, which is said to be tight when it has compact level sets. When ‘‘closed’’ is replaced by ‘‘compact’’ in (2), we say that a vague large deviation principle holds. The net ðma Þ is exponentially tight with respect to ðta Þ if for each 40 there exists a compact K X such that lim sup mtaa ðX nKÞo. For each ½1; þ1½-valued Borel measurable function h on X, each aX0 and each 40, we set Gha; ¼ fx 2 X : a oehðxÞ oa þ g and F ha; ¼ fx 2 X : a pehðxÞ pa þ g (we simply write G a; , F a; when a ¼ ehðzÞ for some z 2 X ). When X is a real Hausdorff topological vector space, we denote by X its topological dual endowed with the weak -topology. For each map f : X 7!½1; þ1, we define a map f on X by f ðlÞ ¼ supx2X flðxÞ f ðxÞg for all l 2 X . By applying this definition to f , one obtains a map f on X defined by f ðxÞ ¼ supl2X flðxÞ f ðlÞg for all x 2 X . It is easy to see that f coincides with the affine regularization of f (i.e., the pointwise supremum of the continuous affine functions everywhere less than f). When moreover X is locally convex and f is convex lower semi-continuous with the property that f 1 when f takes the value 1, then f ¼ f (Ekeland and Teman, 1976, Proposition 4.1). 2. Variational representation for LðÞ Under exponential tightness hypothesis, part (b) of the following theorem gives a necessary and sufficient condition in order that LðhÞ have the same form as when large deviations hold (see Comman, 2003, for a general version of Varadhan’s theorem valid on any topological space and without tightness hypothesis). This condition is strictly weaker than the usual Varadhan’s tail condition, since this one requires that the LHS of (5) vanishes. Theorem 1. Let X be a topological space in which compact sets are Borel sets, let h 2 CðX Þ, and assume that either h 2 CK ðX Þ or ðma Þ is exponentially tight with respect to ðta Þ. The following conclusions hold: (a) For each real M, we have sup x2fhoMg
ehðxÞl 0 ðxÞ p lim sup mtaa ðeh=ta 1fhpMg Þp sup
ehðxÞl 0 ðxÞ .
(3)
x2fhpMg
In particular, lim lim sup mtaa ðeh=ta 1fhpMg Þ ¼ sup ehðxÞl 0 ðxÞ .
M!þ1
x2X
(4)
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(b) LðhÞ ¼ supx2X fhðxÞ l 0 ðxÞg if and only if lim lim sup mtaa ðeh=ta 1fh4Mg Þp lim lim sup mtaa ðeh=ta 1fhpMg Þ.
M!þ1
(5)
M!þ1
If the above inequality is strict, then for each real M large enough, we have sup fhðxÞ l 0 ðxÞg.
LðhÞ ¼
(6)
x2fhpMg
Proof. For each real M, we put hM ¼ h1fhpMg 11fh4Mg (with the convention ‘‘1 0 ¼ 0’’). By Theorem 3.1 of Comman (2003) applied to hM , we get lim sup mtaa ðehM =ta 1fhpMg Þ ¼
fðehM ðxÞ Þ lim sup mtaa ðF ehM ðxÞ ; \ fhpMgÞg.
sup
x2fhpMg;40
Since hM coincides with h on fhpMg, we have F ehM ðxÞ ; \ fhpMg ¼ F ehðxÞ ; \ fhpMg, hence lim sup mtaa ðeh=ta 1fhpMg Þ ¼
sup x2fhpMg;40
sup
X
x2fhoMg;40
fðehðxÞ Þ lim sup mtaa ðF ehðxÞ ; \ fhpMgÞg fðehðxÞ Þ lim sup mtaa ðG ehðxÞ ; \ fhoMgÞgX sup
ehðxÞl 0 ðxÞ ,
ð7Þ
x2fhoMg
which proves the first inequality of (3). Since the second inequality of (3) holds obviously when lim sup mtaa ðeh=ta 1fhpMg Þ ¼ 0, we assume that lim sup mtaa ðeh 1fhpMg Þ4em for some real m. Suppose that sup ehðxÞl 0 ðxÞ þ no lim sup mtaa ðeh=ta 1fhpMg Þ
x2fhpMg
for some n40. First assume that exponential tightness holds, and let K X be a compact set such that lim sup mtaa ðX nKÞoemM , and get by (7) lim sup mtaa ðeh=ta 1fhpMg Þ ¼
fðehðxÞ Þ lim sup mtaa ðF ehðxÞ ; \ fhpMg \ KÞg.
sup
x2fhpMg;40
Then, there exists x0 2 fhpMg and 0 40 with ehðx0 Þ 40 such that sup x2fhpMg
ehðxÞl 0 ðxÞ oðehðx0 Þ 0 nÞ lim sup mtaa ðF ehðx0 Þ ; \ fhpMg \ KÞ.
(8)
0
For each x 2 F ehðx0 Þ ; \ fhpMg \ K, there is an open set V x containing x, and such that ehðyÞ 4ehðx0 Þ 0 n 0
for all y 2 V x . By (8), for each x 2 F ehðx0 Þ ; \ fhpMg \ K, there is an open set W x containing x, and such that 0
e
hðxÞ
lim
sup mtaa ðW x Þoðehðx0 Þ
0 nÞ lim sup mtaa ðF ehðx0 Þ ; \ fhpMg \ KÞ.
(9)
0
Put G x ¼ W x \ V x for all x 2 F ehðx0 Þ ; \ fhpMg \ K. Since F ehðx0 Þ ; \ fhpMg \ K is compact, there is a finite 0 0S set A F ehðx0 Þ ; \ fhpMg \ K such that F ehðx0 Þ ; \ fhpMg \ K x2A G x , hence for some x 2 A, 0
ðe
hðx0 Þ
0 nÞ lim
0
sup mtaa ðF ehðx0 Þ ; 0
\ fhpMg \ KÞpehðxÞ lim sup mtaa ðG x Þ,
(10)
which contradicts (9); therefore, lim sup mtaa ðeh=ta 1fhpMg Þpsupx2fhpMg ehðxÞl 0 ðxÞ and (3) holds. If h 2 CK ðX Þ, then the above proof works verbatim with F ehðx0 Þ ; \ fhpMg in place of F ehðx0 Þ ; \ fhpMg \ K in (8)–(10), 0
0
since F ehðx0 Þ ; \ fhpMg is compact. Then, (4) is a direct consequence of (3), and so (a) holds. The first 0
assertion of (b) follows from (4) and the fact that for each real M, lim sup mtaa ðeh=ta Þ ¼ lim sup mtaa ðeh=ta 1fhpMg Þ _ lim sup mtaa ðeh=ta 1fh4Mg Þ. If the inequality in (5) is strict, then lim sup mtaa ðeh=ta Þ ¼ lim sup mtaa ðeh=ta 1fhpMg Þ for all M large enough, and (6) follows from (a). & The following corollary shows that under exponential tightness, and for a given h 2 CðX Þ, all the conclusions of Varadhan’s theorem remain true with l 0 in place of the rate function, and replacing the large deviation hypothesis by the weaker condition (11); the existence of LðhÞ follows from Corollary 1 of Comman
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(2005) by noting that its conclusion still holds when the usual tail condition is replaced by a strict inequality in (5). Note that F ha;d is compact when h 2 CK ðX Þ and a d40, so that (11) is weaker than vague large deviations. Corollary 1. Let X be a topological space in which compact sets are Borel sets, let h 2 CðX Þ satisfying (5) strictly and lim sup mtaa ðF ha;d Þp lim inf mtaa ðG ha; Þ
(11)
for all reals a ^ 4d40. If either ðma Þ is exponentially tight with respect to ðta Þ or h 2 CK ðX Þ, then LðhÞ exists and for each real M large enough, LðhÞ ¼
sup fhðxÞ l 0 ðxÞg ¼ supfhðxÞ l 0 ðxÞg. x2fhpMg
x2X
3. The affine regularization of a rate function In this section, X is a real Hausdorff topological vector space, and X denotes its topological dual. Let us consider the maps L~ and L defined on X , respectively, by ~ LðlÞ ¼ lim lim sup ta log ma ðel=ta 1flpMg Þ M!þ1
and LðlÞ ¼ lim sup ta log ma ðel=ta Þ for all l 2 X ; we write LðlÞ when the limit exists in the last expression (L is then the usual generalized log~ moment generating function). Note that Lð0Þ ¼ 0, and L~ is convex since the map l7! log ma ðel=ta 1fhpMg Þ is convex by Ho¨lder’s inequality. If X ¼ R or ðma Þ is exponentially tight with respect to ðta Þ, then L~ is weak lower semi-continuous by (4), which proves part (a) of Corollary 2. Part (c) follows from (a), (b) and the fact that the finiteness of L implies the Varadhan’s tail condition for all l 2 X (Dembo and Zeitouni, 1998, Lemma 4.3.8); (c) was known under the hypotheses that X is locally convex and large deviations hold with tight rate function (Dembo and Zeitouni, 1998, Remark, p. 153). Corollary 2. Let X be a real Hausdorff topological vector space. If X ¼ R or if ðma Þ is exponentially tight with respect to ðta Þ, then
(a) (b) (c)
L~ ¼ l 0 ; in particular, L~ is weak lower semi-continuous. ~ For each l 2 X , LðlÞ ¼ LðlÞ if and only if (5) holds with h ¼ l. L is weak lower semi-continuous when it is finite-valued.
We assume now that ðma Þ satisfies a large deviation principle with powers ðta Þ and rate function J. When L does not take the value þ1, each l 2 X satisfies the tail condition of Varadhan’s theorem, hence by the tightness-free version of this theorem (Comman, 2003, Corollary 3.4), LðlÞ exists for all l 2 X and L ¼ J , and we get the following result, which is usually stated in the literature under the extra hypothesis that J is tight (see Deuschel and Strook, 1989, Theorem 2.2.21; Dembo and Zeitouni, 1998, Theorem 4.5.10):
L is the affine regularization of J, and J is convex if and only if J ¼ L when moreover X is locally convex.
When L takes the value þ1, the Varadhan’s theorem cannot be applied in order to get the existence of LðlÞ. However, when exponential tightness holds we can use Corollary 1 to get a strictly weaker condition than the finiteness of L in order that the above result remains true; with a bit weaker condition it suffices to replace L by L; this is the content of part (b) of Corollary 3. Part (a) shows that in any case it holds verbatim replacing L ~ Moreover, it suffices to assume vague large deviations when X ¼ R; in particular any convex rate by L. function governing a vague large deviation principle is given by L~ .
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Corollary 3. We assume that one of the following conditions holds: (i) X ¼ R and ðma Þ satisfies a vague large deviation principle with powers ðta Þ and rate function J. (ii) X is a real Hausdorff topological vector space, ðma Þ is exponentially tight with respect to ðta Þ, and ðma Þ satisfies a large deviation principle with powers ðta Þ and rate function J. Then, (a) L~ is the affine regularization of J. When moreover X is locally convex, J is convex if and only if J ¼ L~ . ~ (b) If (5) holds (resp. holds strictly) for all l 2 X , then (a) holds verbatim with L (resp. L) in place of L.
Proof. Since J ¼ l 0 we get L~ ¼ J by Corollary 2(a), which proves the first assertion of (a); the second one follows from local convexity. The first assertion of (b) follows from Corollary 2(b); the existence of LðlÞ in the second assertion follows from Corollary 1. & Example 1. Consider the net ðm Þ40 , where m is the probability measure on R defined by m ð0Þ ¼ 1 2p , m ð log p Þ ¼ m ð log p Þ ¼ p , and assume that lim log p ¼ 1. It is easy to see that ðm Þ40 satisfies a large deviation principle with the convex rate function ( 0 if x ¼ 0; JðxÞ ¼ þ1 otherwise: For each real l, we have LðlÞ ¼ lim ðl þ 1Þ log p _ lim logð1 2p Þ _ lim ð1 lÞ log p , hence
(
LðlÞ ¼
0 þ1
if jljp1; ifjlj41
and L ðxÞ ¼ jxj for all x 2 R. On the other hand, we have ( lim ðl þ 1Þ log p _ lim logð1 2p Þ ¼ 0 if lX0; ~ LðlÞ ¼ lim logð1 2p Þ _ lim ð1 lÞ log p ¼ 0 if lo0
and so J ¼ L~ aL . We give now a slight strengthening of Baldi’s theorem by asking only that (5) holds strictly for the exposed hyperplanes l for which LðlÞ exists, in place of the stronger hypothesis LðclÞo1 for some c41 (which implies the Varadhan’s tail condition for l, and which is actually used in the usual proofs). Note that our proof is very short and direct in comparison. Theorem 2. Let X be a real Hausdorff topological vector space and assume that ðma Þ is exponentially tight with respect to ðta Þ. Let E be the set of exposed points x of L for which there is an exposed hyperplane lx such that Lðlx Þ exists and (5) holds strictly. If inf G L ¼ inf G\E L for all open G X , then ðma Þ satisfies a large deviation principle with powers ðta Þ and rate function L . Proof. Let d4supx2K eL such that
ðxÞ
for some compact K X and some real d. For each x 2 K there exists lx 2 X
d4eLðlx Þlx x Xelx x sup elx zl 0 ðzÞ , z2X
where the last inequality follows from (4), and by taking z ¼ x we get d4el 0 ðxÞ and so d4 lim sup mtaa ðG x Þ for some open G x containing x. Since K is compact it can be covered by a finite number of such G x , which gives d4 lim sup mtaa ðKÞ; this proves the upper bounds since K and d are arbitrary. Let x 2 E with some exposed t hyperplane lx for which Lðlx Þ exists and (5) holds strictly. Let ðmbb Þ be a subnet of ðmtaa Þ. Since X is regular, by
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exponential tightness ðmb Þ has a subnet ðmg Þ satisfying a large deviation principle with powers ðtg Þ and rate t
t
ðm g Þ
ðm g Þ
t
function l 0 g (where l 0 g is the function obtained replacing mtaa by mgg in the definition of l 0 ). By Theorem 1 t
ðm g Þ
t
applied to the net ðmgg Þ, and since Lðlx Þ exists, there is x0 2 X such that Lðlx Þ ¼ lx x0 l 0 g ðx0 Þ. We have t
ðm g Þ
lx x0 L ðx0 ÞXlx x0 l 0 ðx0 ÞXlx x0 l 0 g ðx0 Þ ¼ Lðlx ÞXL ðlx Þ ¼ lx x L ðxÞ,
where the first inequality follows by noting that L pl 0 since l 0 pL by (4). Since x is exposed with exposed t
ðm g Þ
hyperplane lx , all the above inequalities are equalities and x ¼ x0 hence l 0 g ðxÞ ¼ L ðxÞ. Since large deviation t ðmgg Þ
t
hold for ðmgg Þ, we have l 0 t ðmgg Þ
t ðmgg Þ
¼ l1
t ðmgg Þ
(where l 1 t
l 1 ðxÞ ¼ L ðxÞ. Therefore, lim inf mgg ðGÞXeL in E, we get t
lim inf mgg ðGÞX sup eL x2G\E
ðxÞ
¼ sup eL
ðxÞ
t
ðxÞ ¼ log infflim inf mgg ðGÞ : x 2 G X ; G openg), so that
ðxÞ
for all open G containing x, and since x is arbitrary
.
x2G t
Since the upper bounds hold obviously for ðmgg Þ, we have proved that any subnet of ðmtaa Þ has a subnet satisfying a large deviation principle with rate function L . Seeing a large deviation principle as a convergence in a suitable space of set-functions (see Comman, 2003, Remark 3.5), we get the same result for ðmtaa Þ. & References Comman, H., 2005. Functional approach of large deviations in general spaces. J. Theoret. Probab. 18 (1), 187–207. Comman, H., 2003. Criteria for large deviations. Trans. Amer. Math. Soc. 355, 2905–2923. Dembo, A., Zeitouni, O., 1998. Large Deviations Techniques and Applications, second ed. Springer, New York. Deuschel, J.D., Strook, D.W., 1989. Large Deviations. Academic Press, San Diego. Ekeland, I., Teman, T., 1976. Convex Analysis and Variational Problems. North-Holland, Amsterdam.
Statistics & Probability Letters 77 (2007) 931–936 www.elsevier.com/locate/stapro
Variational form of the large deviation functional$ Henri Comman Department of Mathematics, University of Santiago de Chile, Bernardo O’Higgins 3363, Santiago, Chile Received 16 August 2005; received in revised form 27 January 2006; accepted 17 January 2007 Available online 9 February 2007
Abstract Under exponential tightness hypothesis, we show that the large deviation functional LðÞ has the same variational form as when large deviations hold. As an application, we identify the affine regularization of a rate function on any real Hausdorff topological vector space in terms of LðÞ. Some classical results involving the generalized log-moment generating function are improved. r 2007 Elsevier B.V. All rights reserved. Keywords: Large deviations; Generalized log-moment generating function
1. Introduction Let ðma Þ be a net of Borel probability measures on a topological space X, and let ðta Þ be a net in 0; þ1½ converging function h on X, we write mtaa ðeh=ta Þ for R hðxÞ=t to 0. taFor any ½1; þ1½-valued Borel tmeasurable h=ta a a ð X e ma ðdxÞÞ , and define LðhÞ ¼ log lim sup ma ðe Þ, and LðhÞ ¼ log lim mtaa ðeh=ta Þ when this limit exists. We denote by CðX Þ the set of ½1; þ1½-valued continuous functions on X. The usual version of Varadhan’s theorem for X Polish asserts that under exponential tightness hypothesis, a large deviation principle for ðmtaa Þ with rate function J implies the existence of LðhÞ for all h 2 CðX Þ satisfying some tail condition, with LðhÞ ¼ supfhðxÞ JðxÞg.
(1)
x2X
In this note, we show that if the large deviation hypothesis is dropped, then (1) remains true with LðhÞ in place of LðhÞ, and J replaced by the function l 0 , where l 0 ðxÞ ¼ log infflim sup mtaa ðGÞ : x 2 G X ; G openg for all x 2 X (note that a large deviation principle on a regular space is always governed by l 0 ); furthermore, this is valid for any topological space in which compact sets are Borel sets (Theorem 1). If moreover the exponential tightness is dropped, then the result remains true replacing CðX Þ by the set CK ðX Þ of elements h 2 CðX Þ for which fy 2 X : ehðxÞ pehðyÞ pehðxÞ þ g is compact for all x 2 X and 40 with ehðxÞ 4. A general version of Varadhan’s theorem is a direct consequence (Corollary 1). $
This work was supported in part by USACH-DICYT Grant 040533C. E-mail address: [email protected].
0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2007.01.005
ARTICLE IN PRESS H. Comman / Statistics & Probability Letters 77 (2007) 931–936
932
In the last section we apply Theorem 1 to the case where X is a real Hausdorff topological vector space and h belongs to its topological dual X . Assuming the exponential tightness and large deviations for ðmtaa Þ with rate function J, we show that the affine regularization of J is the Legendre–Fenchel transform L~ of the map L~ ~ defined on X by LðlÞ ¼ limM!þ1 Lðl1flpMg 11fl4Mg Þ. It follows that J ¼ L~ when J is convex and X locally convex. We show that some classical results (and in particular the equality J ¼ L or even J ¼ L ) known when Lo1 are still valid under weaker conditions (Corollaries 2, 3); in particular, Baldi’s theorem is slightly improved (Theorem 2). 1.1. Notations We recall that ðma Þ satisfies a large deviation principle with powers ðta Þ if there exists a ½0; þ1-valued lower semi-continuous function J on X such that lim sup mtaa ðF Þp sup eJðxÞ
for all closed F X
sup eJðxÞ p lim inf mtaa ðGÞ
for all open G X ;
(2)
x2F
and x2G
J is called a rate function for ðmtaa Þ, which is said to be tight when it has compact level sets. When ‘‘closed’’ is replaced by ‘‘compact’’ in (2), we say that a vague large deviation principle holds. The net ðma Þ is exponentially tight with respect to ðta Þ if for each 40 there exists a compact K X such that lim sup mtaa ðX nKÞo. For each ½1; þ1½-valued Borel measurable function h on X, each aX0 and each 40, we set Gha; ¼ fx 2 X : a oehðxÞ oa þ g and F ha; ¼ fx 2 X : a pehðxÞ pa þ g (we simply write G a; , F a; when a ¼ ehðzÞ for some z 2 X ). When X is a real Hausdorff topological vector space, we denote by X its topological dual endowed with the weak -topology. For each map f : X 7!½1; þ1, we define a map f on X by f ðlÞ ¼ supx2X flðxÞ f ðxÞg for all l 2 X . By applying this definition to f , one obtains a map f on X defined by f ðxÞ ¼ supl2X flðxÞ f ðlÞg for all x 2 X . It is easy to see that f coincides with the affine regularization of f (i.e., the pointwise supremum of the continuous affine functions everywhere less than f). When moreover X is locally convex and f is convex lower semi-continuous with the property that f 1 when f takes the value 1, then f ¼ f (Ekeland and Teman, 1976, Proposition 4.1). 2. Variational representation for LðÞ Under exponential tightness hypothesis, part (b) of the following theorem gives a necessary and sufficient condition in order that LðhÞ have the same form as when large deviations hold (see Comman, 2003, for a general version of Varadhan’s theorem valid on any topological space and without tightness hypothesis). This condition is strictly weaker than the usual Varadhan’s tail condition, since this one requires that the LHS of (5) vanishes. Theorem 1. Let X be a topological space in which compact sets are Borel sets, let h 2 CðX Þ, and assume that either h 2 CK ðX Þ or ðma Þ is exponentially tight with respect to ðta Þ. The following conclusions hold: (a) For each real M, we have sup x2fhoMg
ehðxÞl 0 ðxÞ p lim sup mtaa ðeh=ta 1fhpMg Þp sup
ehðxÞl 0 ðxÞ .
(3)
x2fhpMg
In particular, lim lim sup mtaa ðeh=ta 1fhpMg Þ ¼ sup ehðxÞl 0 ðxÞ .
M!þ1
x2X
(4)
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(b) LðhÞ ¼ supx2X fhðxÞ l 0 ðxÞg if and only if lim lim sup mtaa ðeh=ta 1fh4Mg Þp lim lim sup mtaa ðeh=ta 1fhpMg Þ.
M!þ1
(5)
M!þ1
If the above inequality is strict, then for each real M large enough, we have sup fhðxÞ l 0 ðxÞg.
LðhÞ ¼
(6)
x2fhpMg
Proof. For each real M, we put hM ¼ h1fhpMg 11fh4Mg (with the convention ‘‘1 0 ¼ 0’’). By Theorem 3.1 of Comman (2003) applied to hM , we get lim sup mtaa ðehM =ta 1fhpMg Þ ¼
fðehM ðxÞ Þ lim sup mtaa ðF ehM ðxÞ ; \ fhpMgÞg.
sup
x2fhpMg;40
Since hM coincides with h on fhpMg, we have F ehM ðxÞ ; \ fhpMg ¼ F ehðxÞ ; \ fhpMg, hence lim sup mtaa ðeh=ta 1fhpMg Þ ¼
sup x2fhpMg;40
sup
X
x2fhoMg;40
fðehðxÞ Þ lim sup mtaa ðF ehðxÞ ; \ fhpMgÞg fðehðxÞ Þ lim sup mtaa ðG ehðxÞ ; \ fhoMgÞgX sup
ehðxÞl 0 ðxÞ ,
ð7Þ
x2fhoMg
which proves the first inequality of (3). Since the second inequality of (3) holds obviously when lim sup mtaa ðeh=ta 1fhpMg Þ ¼ 0, we assume that lim sup mtaa ðeh 1fhpMg Þ4em for some real m. Suppose that sup ehðxÞl 0 ðxÞ þ no lim sup mtaa ðeh=ta 1fhpMg Þ
x2fhpMg
for some n40. First assume that exponential tightness holds, and let K X be a compact set such that lim sup mtaa ðX nKÞoemM , and get by (7) lim sup mtaa ðeh=ta 1fhpMg Þ ¼
fðehðxÞ Þ lim sup mtaa ðF ehðxÞ ; \ fhpMg \ KÞg.
sup
x2fhpMg;40
Then, there exists x0 2 fhpMg and 0 40 with ehðx0 Þ 40 such that sup x2fhpMg
ehðxÞl 0 ðxÞ oðehðx0 Þ 0 nÞ lim sup mtaa ðF ehðx0 Þ ; \ fhpMg \ KÞ.
(8)
0
For each x 2 F ehðx0 Þ ; \ fhpMg \ K, there is an open set V x containing x, and such that ehðyÞ 4ehðx0 Þ 0 n 0
for all y 2 V x . By (8), for each x 2 F ehðx0 Þ ; \ fhpMg \ K, there is an open set W x containing x, and such that 0
e
hðxÞ
lim
sup mtaa ðW x Þoðehðx0 Þ
0 nÞ lim sup mtaa ðF ehðx0 Þ ; \ fhpMg \ KÞ.
(9)
0
Put G x ¼ W x \ V x for all x 2 F ehðx0 Þ ; \ fhpMg \ K. Since F ehðx0 Þ ; \ fhpMg \ K is compact, there is a finite 0 0S set A F ehðx0 Þ ; \ fhpMg \ K such that F ehðx0 Þ ; \ fhpMg \ K x2A G x , hence for some x 2 A, 0
ðe
hðx0 Þ
0 nÞ lim
0
sup mtaa ðF ehðx0 Þ ; 0
\ fhpMg \ KÞpehðxÞ lim sup mtaa ðG x Þ,
(10)
which contradicts (9); therefore, lim sup mtaa ðeh=ta 1fhpMg Þpsupx2fhpMg ehðxÞl 0 ðxÞ and (3) holds. If h 2 CK ðX Þ, then the above proof works verbatim with F ehðx0 Þ ; \ fhpMg in place of F ehðx0 Þ ; \ fhpMg \ K in (8)–(10), 0
0
since F ehðx0 Þ ; \ fhpMg is compact. Then, (4) is a direct consequence of (3), and so (a) holds. The first 0
assertion of (b) follows from (4) and the fact that for each real M, lim sup mtaa ðeh=ta Þ ¼ lim sup mtaa ðeh=ta 1fhpMg Þ _ lim sup mtaa ðeh=ta 1fh4Mg Þ. If the inequality in (5) is strict, then lim sup mtaa ðeh=ta Þ ¼ lim sup mtaa ðeh=ta 1fhpMg Þ for all M large enough, and (6) follows from (a). & The following corollary shows that under exponential tightness, and for a given h 2 CðX Þ, all the conclusions of Varadhan’s theorem remain true with l 0 in place of the rate function, and replacing the large deviation hypothesis by the weaker condition (11); the existence of LðhÞ follows from Corollary 1 of Comman
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(2005) by noting that its conclusion still holds when the usual tail condition is replaced by a strict inequality in (5). Note that F ha;d is compact when h 2 CK ðX Þ and a d40, so that (11) is weaker than vague large deviations. Corollary 1. Let X be a topological space in which compact sets are Borel sets, let h 2 CðX Þ satisfying (5) strictly and lim sup mtaa ðF ha;d Þp lim inf mtaa ðG ha; Þ
(11)
for all reals a ^ 4d40. If either ðma Þ is exponentially tight with respect to ðta Þ or h 2 CK ðX Þ, then LðhÞ exists and for each real M large enough, LðhÞ ¼
sup fhðxÞ l 0 ðxÞg ¼ supfhðxÞ l 0 ðxÞg. x2fhpMg
x2X
3. The affine regularization of a rate function In this section, X is a real Hausdorff topological vector space, and X denotes its topological dual. Let us consider the maps L~ and L defined on X , respectively, by ~ LðlÞ ¼ lim lim sup ta log ma ðel=ta 1flpMg Þ M!þ1
and LðlÞ ¼ lim sup ta log ma ðel=ta Þ for all l 2 X ; we write LðlÞ when the limit exists in the last expression (L is then the usual generalized log~ moment generating function). Note that Lð0Þ ¼ 0, and L~ is convex since the map l7! log ma ðel=ta 1fhpMg Þ is convex by Ho¨lder’s inequality. If X ¼ R or ðma Þ is exponentially tight with respect to ðta Þ, then L~ is weak lower semi-continuous by (4), which proves part (a) of Corollary 2. Part (c) follows from (a), (b) and the fact that the finiteness of L implies the Varadhan’s tail condition for all l 2 X (Dembo and Zeitouni, 1998, Lemma 4.3.8); (c) was known under the hypotheses that X is locally convex and large deviations hold with tight rate function (Dembo and Zeitouni, 1998, Remark, p. 153). Corollary 2. Let X be a real Hausdorff topological vector space. If X ¼ R or if ðma Þ is exponentially tight with respect to ðta Þ, then
(a) (b) (c)
L~ ¼ l 0 ; in particular, L~ is weak lower semi-continuous. ~ For each l 2 X , LðlÞ ¼ LðlÞ if and only if (5) holds with h ¼ l. L is weak lower semi-continuous when it is finite-valued.
We assume now that ðma Þ satisfies a large deviation principle with powers ðta Þ and rate function J. When L does not take the value þ1, each l 2 X satisfies the tail condition of Varadhan’s theorem, hence by the tightness-free version of this theorem (Comman, 2003, Corollary 3.4), LðlÞ exists for all l 2 X and L ¼ J , and we get the following result, which is usually stated in the literature under the extra hypothesis that J is tight (see Deuschel and Strook, 1989, Theorem 2.2.21; Dembo and Zeitouni, 1998, Theorem 4.5.10):
L is the affine regularization of J, and J is convex if and only if J ¼ L when moreover X is locally convex.
When L takes the value þ1, the Varadhan’s theorem cannot be applied in order to get the existence of LðlÞ. However, when exponential tightness holds we can use Corollary 1 to get a strictly weaker condition than the finiteness of L in order that the above result remains true; with a bit weaker condition it suffices to replace L by L; this is the content of part (b) of Corollary 3. Part (a) shows that in any case it holds verbatim replacing L ~ Moreover, it suffices to assume vague large deviations when X ¼ R; in particular any convex rate by L. function governing a vague large deviation principle is given by L~ .
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Corollary 3. We assume that one of the following conditions holds: (i) X ¼ R and ðma Þ satisfies a vague large deviation principle with powers ðta Þ and rate function J. (ii) X is a real Hausdorff topological vector space, ðma Þ is exponentially tight with respect to ðta Þ, and ðma Þ satisfies a large deviation principle with powers ðta Þ and rate function J. Then, (a) L~ is the affine regularization of J. When moreover X is locally convex, J is convex if and only if J ¼ L~ . ~ (b) If (5) holds (resp. holds strictly) for all l 2 X , then (a) holds verbatim with L (resp. L) in place of L.
Proof. Since J ¼ l 0 we get L~ ¼ J by Corollary 2(a), which proves the first assertion of (a); the second one follows from local convexity. The first assertion of (b) follows from Corollary 2(b); the existence of LðlÞ in the second assertion follows from Corollary 1. & Example 1. Consider the net ðm Þ40 , where m is the probability measure on R defined by m ð0Þ ¼ 1 2p , m ð log p Þ ¼ m ð log p Þ ¼ p , and assume that lim log p ¼ 1. It is easy to see that ðm Þ40 satisfies a large deviation principle with the convex rate function ( 0 if x ¼ 0; JðxÞ ¼ þ1 otherwise: For each real l, we have LðlÞ ¼ lim ðl þ 1Þ log p _ lim logð1 2p Þ _ lim ð1 lÞ log p , hence
(
LðlÞ ¼
0 þ1
if jljp1; ifjlj41
and L ðxÞ ¼ jxj for all x 2 R. On the other hand, we have ( lim ðl þ 1Þ log p _ lim logð1 2p Þ ¼ 0 if lX0; ~ LðlÞ ¼ lim logð1 2p Þ _ lim ð1 lÞ log p ¼ 0 if lo0
and so J ¼ L~ aL . We give now a slight strengthening of Baldi’s theorem by asking only that (5) holds strictly for the exposed hyperplanes l for which LðlÞ exists, in place of the stronger hypothesis LðclÞo1 for some c41 (which implies the Varadhan’s tail condition for l, and which is actually used in the usual proofs). Note that our proof is very short and direct in comparison. Theorem 2. Let X be a real Hausdorff topological vector space and assume that ðma Þ is exponentially tight with respect to ðta Þ. Let E be the set of exposed points x of L for which there is an exposed hyperplane lx such that Lðlx Þ exists and (5) holds strictly. If inf G L ¼ inf G\E L for all open G X , then ðma Þ satisfies a large deviation principle with powers ðta Þ and rate function L . Proof. Let d4supx2K eL such that
ðxÞ
for some compact K X and some real d. For each x 2 K there exists lx 2 X
d4eLðlx Þlx x Xelx x sup elx zl 0 ðzÞ , z2X
where the last inequality follows from (4), and by taking z ¼ x we get d4el 0 ðxÞ and so d4 lim sup mtaa ðG x Þ for some open G x containing x. Since K is compact it can be covered by a finite number of such G x , which gives d4 lim sup mtaa ðKÞ; this proves the upper bounds since K and d are arbitrary. Let x 2 E with some exposed t hyperplane lx for which Lðlx Þ exists and (5) holds strictly. Let ðmbb Þ be a subnet of ðmtaa Þ. Since X is regular, by
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exponential tightness ðmb Þ has a subnet ðmg Þ satisfying a large deviation principle with powers ðtg Þ and rate t
t
ðm g Þ
ðm g Þ
t
function l 0 g (where l 0 g is the function obtained replacing mtaa by mgg in the definition of l 0 ). By Theorem 1 t
ðm g Þ
t
applied to the net ðmgg Þ, and since Lðlx Þ exists, there is x0 2 X such that Lðlx Þ ¼ lx x0 l 0 g ðx0 Þ. We have t
ðm g Þ
lx x0 L ðx0 ÞXlx x0 l 0 ðx0 ÞXlx x0 l 0 g ðx0 Þ ¼ Lðlx ÞXL ðlx Þ ¼ lx x L ðxÞ,
where the first inequality follows by noting that L pl 0 since l 0 pL by (4). Since x is exposed with exposed t
ðm g Þ
hyperplane lx , all the above inequalities are equalities and x ¼ x0 hence l 0 g ðxÞ ¼ L ðxÞ. Since large deviation t ðmgg Þ
t
hold for ðmgg Þ, we have l 0 t ðmgg Þ
t ðmgg Þ
¼ l1
t ðmgg Þ
(where l 1 t
l 1 ðxÞ ¼ L ðxÞ. Therefore, lim inf mgg ðGÞXeL in E, we get t
lim inf mgg ðGÞX sup eL x2G\E
ðxÞ
¼ sup eL
ðxÞ
t
ðxÞ ¼ log infflim inf mgg ðGÞ : x 2 G X ; G openg), so that
ðxÞ
for all open G containing x, and since x is arbitrary
.
x2G t
Since the upper bounds hold obviously for ðmgg Þ, we have proved that any subnet of ðmtaa Þ has a subnet satisfying a large deviation principle with rate function L . Seeing a large deviation principle as a convergence in a suitable space of set-functions (see Comman, 2003, Remark 3.5), we get the same result for ðmtaa Þ. & References Comman, H., 2005. Functional approach of large deviations in general spaces. J. Theoret. Probab. 18 (1), 187–207. Comman, H., 2003. Criteria for large deviations. Trans. Amer. Math. Soc. 355, 2905–2923. Dembo, A., Zeitouni, O., 1998. Large Deviations Techniques and Applications, second ed. Springer, New York. Deuschel, J.D., Strook, D.W., 1989. Large Deviations. Academic Press, San Diego. Ekeland, I., Teman, T., 1976. Convex Analysis and Variational Problems. North-Holland, Amsterdam.