Stone-Weierstrass type theorems for large deviations

We give a general version of Bryc's theorem valid on any topological space and with any algebra $\mathcal{A}$ of real-valued continuous functions separating the points, or any well-separating class. In absence of exponential tightness, and when the underlying space is locally compact regular and $\mathcal{A}$ constituted by functions vanishing at infinity, we give a sufficient condition on the functional $\Lambda(\cdot)_{\mid \mathcal{A}}$ to get large deviations with not necessarily tight rate function. We obtain the general variational form of any rate function on a completely regular space; when either exponential tightness holds or the space is locally compact Hausdorff, we get it in terms of any algebra as above. Prohorov-type theorems are generalized to any space, and when it is locally compact regular the exponential tightness can be replaced by a (strictly weaker) condition on $\Lambda(\cdot)_{\mid \mathcal{A}}$.

We prove here that this theorem is still true replacing C b (X) by the bounded above part A ba of any algebra A of real-valued (not necessarily bounded) continuous functions separating the points, or any well-separating class; the rate function is then given by ∀x ∈ X, J(x) = sup and we can replace in the above expression A by its negative part A − when A does not vanish identically at any point of X (in particular when A contains the constants); this is the most tricky part because of the lack of stability by translation of A − .As a consequence we obtain the variational form of any rate function (under exponential tightness) on such a space in terms of any such a set A; without assuming exponential tightness, we get it only in terms of C(X) and C(X) − (Corollary 1).
In fact, the main result (Theorem 1) gives a general version valid for any topological space X with the restriction that (1) holds only on the "completely regular part" X 0 of X; however, (1) determines completely the rate function when X is regular.Various kinds of hypotheses are proposed, which all reduce to exponential tightness in the completely regular (not necessarily Hausdorff) case; for instance, the simpler one requires that the tightening compact sets be included in X 0 .This allows us to extend some known results by relaxing topological assumptions on the space: Corollary 2 is a Prohorov-type result valid on any topological space, when preceding versions assumed complete regularity and Hausdorffness; Corollary 3 consider the well-separating class constituted by continuous affine functionals on any real Hausdorff topological vector space, when a preceding version assumed metrizability.
When exponential tightness fails, but assuming X locally compact regular (not necessarily Hausdorff) and A an algebra of continuous functions vanishing at infinity which separates the points and does not vanish identically at any point of X, we give a necessary and sufficient condition on Λ(•) |A to get large deviations with a rate function satisfying a property weaker than the tightness, and having still the form (1) (Theorem 2).A similar condition allows to get the large deviation principle for some subnets (Corollary 5).When X is moreover Hausdorff we show that a large deviation principe is always governed by the rate function (1) with any A as above (Corollary 4).This is achieved by applying the results of [4] and [5]; we use in particular the notion of approximating class, that is a set of functions on which the existence of Λ(•) implies large deviations lower bounds with a function satisfying (2).When X is completely regular, the upper bounds on compact sets hold also with the function (2), which turns to be the rate function when large deviations hold; in the general case some extra conditions are required in order that (2) be a rate function on X 0 .Note that in absence of regularity the identification of a rate function is quite difficult by the lack of uniqueness.
More precisely, an approximating class is a set T of [−∞, +∞[-valued continuous functions on X such that for each x ∈ X, each open set G containing x, each real s > 0, and each real t > 0, there exists h ∈ T satisfying In [4] we proved that if Λ(•) exists on T and under some extra condition (namely, (iii) of Proposition 2), then (µ α ) satisfies a large deviation principle with powers (t α ) and a rate function verifying where T T denotes the set of elements in T satisfying the usual tail condition of Varadhan's theorem.We first improve that by showing that the sup in (2) can be taken on T and T − in place of T T , and Λ(h) can be replaced by Λ(h) and Λ(h) (Proposition 2).This result is used in a crucial way in the sequel.Indeed, we first show that the existence of Λ(•) on A ba implies the existence of Λ(•) on C(X) − ; next, we get large deviations with rate function satisfying (1) on X 0 by applying Proposition 2 with the approximating class C(X).The case with well-separating class is proved similarly.The paper is organized as follows.Section 2 gives the general form of a rate function in terms of an approximating class, strengthening a result of [4].In Section 3 we establish the general versions of Bryc's theorem; an example of Hausdorff regular space where the usual form does not work only because of one point, but satisfying our general hypotheses is given.In Section 4 we study the case where X is locally compact regular.
2 General form of a rate function Throughout the paper X denotes a topological space in which compact sets are Borel sets (in particular, no separation axiom is required) and C(X) denotes the set of all real-valued continuous functions on X.We exclude the trivial case where X is reduced to one point.We recall that a set C ⊂ C(X) "separates the points of X" if for any pair of points x = y in X there exists h ∈ C such that h(x) = h(y).By "C does not vanish identically at any point of X" we mean that for any x ∈ X there exists h ∈ C such that h(x) = 0. Note that any well-separating class satisfies the two above properties.Let F , G, K denote respectively the set of closed, open, and compact subsets of X, and let l be a [0, +∞]-valued function on X.We say that (µ α ) satisfies the large deviation upper (resp.lower) bounds with powers (t α ) and function l if lim sup µ tα α (F ) ≤ sup x∈F e −l(x) for all F ∈ F, (resp.sup When (3) (resp.(3) with K in place of F ) and (4) hold, we say that (µ α ) satisfies a large deviation principle (resp.vague large deviation principle) with powers (t α ); in this case, the lower-regularization of l (i.e., the greatest lower semi-continuous function lesser than l) is called a rate function, which is said to be tight when it has compact level sets.As it is well known ([6], Lemma 4.1.4and Remark pp.118), when X is regular a rate function is uniquely determined and coincides with the function l 0 defined by The following proposition will be used in the sequel since we will deal with functions which do not necessarily satisfy the usual tail condition lim It is easy to see that it generalizes Varadhan's theorem; indeed, since for each [−∞, +∞[valued Borel function h on X, for each subnet (µ β (e h/t β 1 {h≥M} ), it follows that when h ∈ C(X) and satisfies (5), by letting M → +∞ Proposition 1 implies lim sup µ t β β (e h/t β ) = sup x∈X e h(x)−l(x) hence the existence of Λ(h) (since this expression does not depend on the subnet along which the upper limit is taken).
Proposition 1 If the large deviation upper (resp.lower) bounds hold with some function l, then for each h ∈ C(X) and each real M we have In particular when the upper and the lower bounds hold with l we have Proof.For each [−∞, +∞[-valued Borel measurable function h on X, each ε > 0, and each x ∈ X, we put G e h(x) ,ε = {y ∈ X : e h(x) − ε < e h(y) < e h(x) + ε} and F e h(x) ,ε = {y ∈ X : e h(x) − ε ≤ e h(y) ≤ e h(x) + ε}.First assume that the large deviation lower bounds hold with l.For each real M , by Theorem 3.1 of [5] applied to the function k = h1 {h<M} − ∞1 {h≥M} (with the convention "∞ • 0 = 0"), there exists a subnet (µ (where the last inequality follows from the large deviations lower bounds), which proves the lower bounds case.Assume now that the large deviation upper bounds hold with l, and suppose that lim sup µ tα α (e h/tα for some real M .Applying Theorem 3.1 of [5] as above with F e h(x) ,ε in place of G e h(x) ,ε yields and therefore there exists x ∈ X and ε > 0 such that By the large deviation upper-bounds we have and so there exists x) .
Since e h(x ′ ) ≥ e h(x) − ε we obtain e h(x ′ ) e −l(x ′ ) > sup x∈X,h(x)≤M e h(x)−l(x) and the contradiction, which proves the upper bounds case.
We recall here the definition of an approximation class, which involves the set X 0 constituted by the points x ∈ X which can be suitably separated by a continuous function from any closed set not containing x.Note that C(X) − is an approximating class for any space X.It is known that X 0 = X if and only if X is completely regular ([4], Proposition 1).At the other extreme, X 0 = ∅ when C(X) is reduced to constants and X is a T 0 space containing more than one point, as it may occur with some regular Hausdorff spaces ( [7]).Note also that the negative part A − of any approximating class A is again an approximating class.
Definition 1 Let X 0 be the set of points x of X such that for each G ∈ G containing x, each real s > 0, and each real t > 0, there exists h ∈ C b (X) such that A class T of [−∞, +∞[-valued continuous functions on X is said to be approximating if for each x ∈ X 0 , each G ∈ G containing x, each real s > 0, and each real t > 0, T contains some function satisfying (6).
We introduce now a strong variant of exponential tightness by requiring that the tightening compact sets be included in X 0 .Of course, it coincides with the usual one in the completely regular case.

Definition 2
The net (µ α ) is X 0 -exponentially tight with respect to (t α ) if for each ε > 0 there exists a compact set K ⊂ X 0 such that lim sup µ tα α (X\K) < ε.
To any approximating class T we associate the function l T defined by where T T denotes the elements h ∈ T satisfying the tail condition (5), and In Theorem 3 of [4] we proved that the existence of Λ(•) on T together with the condition (iii) below imply a large deviation principle with rate function l T ; in fact, it is easy to verify that the existence of Λ(•) on T − together with (iii) are sufficient.The following proposition shows that we can replace T T by T (resp.T − ) and Λ(h) by Λ(h) in (7) for the case x ∈ X 0 .We can even replace Λ(h) and Λ(h) by lim M Λ(h M ), where h M = h1 {h<M} − ∞1 {h≥M} for all h ∈ T and all reals M .This will be used in the next section in order to obtain the expression of the rate function in terms of the whole algebra (or the well-separating class) since this one may contain unbounded functions.
Proposition 2 Consider the following statements: (i) (µ α ) is X 0 -exponentially tight with respect to (t α ); (ii) (µ α ) is exponentially tight with respect to (t α ) and l 0 |X\X0 = +∞; (iii) For all F ∈ F, for all open covers {G i : i ∈ I} of F ∩ X 0 and for all ε > 0, there exists a finite set {G i1 , ..., G iN } ⊂ {G i : i ∈ I} such that The following conclusions hold.
and (iii) holds.If (i) holds, then the finite cover can be obtained from {G i : i ∈ I} and the above expression is still valid.This proves (a).
Assume that (iii) holds and Λ(•) exists on the negative part T − of some approximating class T .By Theorem 3 of [4] (and the comment before Proposition 2), (µ α ) satisfies a large deviation principle with powers (t α ) and rate function l T defined in (7) with moreover for all G ∈ G.
By lower semi-continuity the above expression determines l T and since T G,s = (T − ) G,s , the same reasoning with the approximating class T − yields l T = l T− .Let x ∈ X 0 and ν be a real such that ν > sup t>0 inf {h∈T :h(x)≥−t} lim M Λ(h M ).For each t > 0, there exists h t ∈ T such that h t (x) ≥ −t and ν > Λ(h t ).By Proposition 1 we get and finally ν ≥ −l T (x) by letting t → 0. Since ν is arbitrary, it follows that Remark 1 It is easy to see that l T (x) = +∞ for all x ∈ X\X 0 ([4], Remark 1).Condition (iii) in Proposition 2 implies that lim µ tα α (F ) = 0 for all closed sets F ⊂ X\X 0 .In fact, the proof of Proposition 3 of [4] shows that the large deviation upper (resp.lower) bounds hold also with the function l defined by it follows that under exponential tightness (ii) is equivalent to (iii) when l is lower semicontinuous and X regular.

Main result
In this section we establish our general version of Bryc's theorem, whose usual algebraic statement in the completely regular Hausdorff case is recovered by taking A = C b (X) in Theorem 1; recall that in this case X 0 = X, so that the general hypotheses reduce to exponential tightness, and (9) coincides with (1).The improvement is threefold: first, it allows a general separating algebra (resp.well-separating class) A; secondly, the rate function is obtained in terms of the negative part A − of A when A does not vanish identically at any point of X; finally, the results hold for any topological space, under the stronger hypothesis of X 0 -exponential tightness (or exponential tightness plus some extra conditions), and with the restriction that the usual form of the rate function is obtained only on X 0 .Let us stress that more than the large deviation property itself, the hard part consists in obtaining the rate function in terms of A and A − , respectively.To our knowledge, up to now, in the algebraic case such formulas were known only when A = C b (X) and X completely regular Hausdorff.The proof is heavily based on Proposition 2. For instance, it is thanks to the expression (8) of the rate function together with Lemma 1 that we can write ( 14) and (15) leading to (9).Lemma 1 For each set T ⊂ C(X) and each x ∈ X we have with equalities when T is stable by translations.The same holds with Λ in place of Λ.
Proof.Let δ be a real such that sup For each t > 0 there exists h t ∈ T such that Λ(h t ) < δ and which proves the first inequality; the proof of the second one is similar.The assertion about the equality is clear, as well as the last assertion.
Theorem 1 Assume that (µ α ) is exponentially tight with respect to (t α ), and satisfies one of the conditions of Proposition 2 (in particular under X 0 -exponential tightness).If Λ(•) exists on the bounded-above part A ba of a set A of real-valued continuous functions on X, which is either an algebra separating the points or a well-separating class, then (µ α ) satisfies a large deviation principle with powers (t α ) and rate function J verifying J |X\X0 = +∞ and ∀x ∈ X 0 , J(x) = sup where h M = h1 {h<M} − ∞1 {h≥M} for all h ∈ A and all reals M .When X is regular (9) determines uniquely the rate function by When A does not vanish identically at any point of X (in particular when A contains the constants as in the well-separating case) it is sufficient to assume the existence of Λ(•) on the negative part A − of A and we can replace A ba by A − in (9).
Proof.Let h ∈ C(X) − such that Λ(h) > −s for some s > 0, and put h s = h ∨ −s.First assume that A is an algebra separating the points and put g = √ −h s .Let B be the algebra generated by A ∪ {c} where c is any nonzero constant function, and note that any element g ∈ B has the form g = k + t for some k ∈ A and some constant t (i.e., B = A + R).By the Stone-Weierstrass theorem, there is a net (g i ) i∈I in B converging uniformly on compact sets to g.Put h i = −g 2 i , k i = h i ∨ −2s for all i ∈ I, and note that (h i ) and (k i ) converge uniformly on compact sets to h s .Let K ∈ K such that lim sup µ tα α (X\K) < e −3s .Assume that Λ(•) exists on A ba , and note that this gives the existence of Λ(•) on B − .Since for each i ∈ I, and each subnet (µ β (e ki/t β 1 X\K ), and e −s ≤ lim sup µ tα α (e hs/tα ) = lim sup µ tα α (e hs/tα 1 K ) ∨ lim sup µ tα α (e hs/tα 1 X\K ), it follows that lim µ tα α (e ki/tα 1 K ) exists with e Λ(ki) = lim µ tα α (e ki/tα 1 K ) for all i ∈ I, and lim sup µ tα α (e hs/tα ) = lim sup µ tα α (e hs/tα 1 K ).
The inequalities log µ tα α (e ki/tα 1 K ) − sup combined with (10) and (11) yield α (e hs/tα 1 K ) ≤ log lim inf µ tα α (e hs/tα ) ≤ log lim sup µ tα α (e hs/tα ) = log lim sup µ tα α (e hs/tα 1 K ) ≤ Λ(k i ) + sup and by taking the limit along i, it follows that Λ(h) exists with Since h is arbitrary in C(X) − , Λ(•) exists on C(X) − and inf By Proposition 2 and Lemma 1 with T = C(X) it follows that (µ α ) satisfies a large deviation principle with powers (t α ) and rate function l C(X) taking infinite value outside X 0 (see Remark 1) and satisfying where the fourth equality follows from (13), and the last two equalities follow by noting that Λ(h) = Λ(k) + t when h = k + t with k ∈ A and t ∈ R. Consequently, the above inequalities are equalities, ( 9) holds and the first assertion of the algebraic case is proved.The assertion concerning the regular case follows from Proposition 2. The last assertion follows by noting that when A does not vanish identically at any point of X (in particular when A contains the constants) we can use A in place of B, and it is sufficient to assume the existence of Λ(•) on A − .Assume now that A is a well-separating class.Let A ∨ − be the set constituted by the finite maxima of elements of A − .By Lemma 4.4.9 of [6] (which remains true for any topological space), for each K ∈ K and each ε > 0 there exists h K,ε ∈ A ∨ − such that sup x∈K |h K,ε (x) − h(x)| < ε.The nets (h i ) i∈I and (k i ) i∈I , where k i = h i ∨ −2s and I = {(K, ε) : K ∈ K, ε > 0} (as a product directed set), converge uniformly on compact sets to h.A similar proof as above with A ∨ − in place of B − gives the existence of Λ(•) on C(X) − , and the large deviation principle with rate function l where the last equality follows by noting that Λ(h) = max 1≤j≤r Λ(h j ) when h = r j=1 h j with {h j : 1 ≤ j ≤ r} ⊂ A. Therefore, the above inequality is an equality, which proves the well-separating case.
The following corollary gives the general variational form of any rate function on a completely regular space in terms of C(X); this result seems new in absence of exponential tightness.When X is Polish or locally compact Hausdorff, it gives the general form of any tight rate function in terms of any separating algebra or well-separating class, since in this case the exponential tightness holds.In the locally compact Hausdorff case, we may compare it with a similar result obtained in the next section when the exponential tightness fails (Corollary 4).and and so, (µ α ) satisfies a vague large deviation principle with power (t α ) and rate function (17), since J = Ĵ|X .This proves the first assertion of (a).Assume moreover that (18) holds.By (20) and ( 21 holds.The second assertion of (a) is proved.Assume that X is Hausdorff and (µ α ) satisfies a large deviation principle with powers (t α ).Then, Λ(•) exists on C 0 (X), and (as proved before) (φ[µ α ]) satisfies a large deviation principle with powers (t α ) and rate function Ĵ defined in (20).Therefore, (µ α ) satisfies a vague large deviation principle with power (t α ) and rate function Ĵ|X , and by uniqueness of a vague rate function on a locally compact Hausdorff space, the rate function coincides with Ĵ|X .This proves the first conclusion of (b).Assume that (19) holds, and define the function l on X by l(x) = J(x) Together with ( 22) and (23), this shows that (φ[µ α ]) satisfies a large deviation principle with powers (t α ) and rate function l, hence l = Ĵ and so Ĵ(∞) = sup x∈X J(x), which implies (18) by ( 20) and (21).
The two following corollaries are the analogues of Corollaries 1 and 2 respectively, for locally compact spaces when the tightness fails (exponential or of the rate function).Indeed, assume that (18) holds.The set { Ĵ > sup x∈X Ĵ(x)} is open in X and since {∞} is not open when X is not compact, we have Ĵ(∞) ≤ sup x∈X Ĵ(x); since the converse inequality holds by (18), we have Ĵ(∞) = sup x∈X Ĵ(x), which is equivalent to (25).