On the Continuity of the Cramer Transform

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(1)On the Continuity of the Cramer Transform Marianne Akian. To cite this version: Marianne Akian. On the Continuity of the Cramer Transform. [Research Report] RR-2841, INRIA. 1996. �inria-00073849�. HAL Id: inria-00073849 https://hal.inria.fr/inria-00073849 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. On the continuity of the Cramer transform M. Akian. N ˚ 2841 Mars 1996 ` THEME 4. ISSN 0249-6399. apport de recherche.

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(4) On the continuity of the Cramer transform M. Akian Thème 4 — Simulation et optimisation de systèmes complexes Projet Méta2 Rapport de recherche n ˚ 2841 — Mars 1996 — 19 pages. Abstract: The Cramer transform introduced in large deviations theory sends classical probabilities (resp. finite positive measures) into (min,+) probabilities (resp. finite measures) also called cost measures. We study its continuity when the two spaces of measures are endowed with the weak convergence topology. We prove that the Cramer transform is continuous in the subspace of logconcave measures and show counter examples in the opposite case. Moreover, in finite dimension, the Cramer transform is bicontinuous. Then, logconcave measures may be identified with lower semicontinuous convex functions. Key-words: Max-plus algebra, Cramer transform, Logconcave measure, Logconcave function, weak convergence, Large deviations, Idempotent measure, Fenchel transform.. (Résumé : tsvp) . e-mail : Marianne.Akian@inria.fr. Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) Téléphone : (33 1) 39 63 55 11 – Télécopie : (33 1) 39 63 53 30.

(5) Sur la continuité de la transformée de Cramer Résumé : La transformée de Cramer, introduite en théorie des grandes déviations, envoie les probabilités classiques (resp. les mesures positives finies) dans les probabilités (resp. les mesures finies) (min,+), appelées mesures de coût. Nous e´ tudions sa continuité quand les espaces de mesures de départ et d’arrivée sont munis des topologies de la convergence faible. Nous prouvons que la transformée de Cramer est continue dans le sous espace des mesures logconcaves et donnons des contreexemples dans le complémentaire. En dimension finie, la transformée de Cramer est de plus bicontinue. Les mesures logconcaves peuvent alors eˆ tre identifiées a` des fonctions convexes semi-continues inférieurement. Mots-clé : Algèbre max-plus, Transformée de Cramer, Mesure logconcave, Fonction logconcave, Convergence faible, Grandes déviations, Mesure idempotente, Transformée de Fenchel..

(6) On the continuity of the Cramer transform. 3. Introduction For any probability of is defined by. #'021 Here, "$#%'&)(+ *-,/and .. on the Borel sets of a topological vector space. . def

(7) . . , the Cramer transform . . . denote respectively the Laplace %'and ! &=<?> Athe @ Fenchel transforms, that is 45 3 6!8 , and where denotes the dual space of 2 7 ; 9 : 3 D B E C ' 5 F 3 4 3 . Then is a lower semicontinuous (l.s.c.) convex function on . The Cramer transform was introduced in large deviations theory [11, 7, 14, 15, 27, 13]. Let

(8) GIH be a sequence of i.i.d. random variables with values in and law . If is finite in a neighborhood of zero, the Cramer theorem states that the law of JK?LNMOH MOM LPJQ obeys a large deviation principle with rate function . This result is true if is a finite dimensional vector space but also if it is locally convex and Hausdorff and if the support of is a subset of a closed convex Polish subspace of [8]. Another domain where the Cramer transform appears to be useful is decision theory. Following @WV the theory of idempotent measures and integrals of Maslov [18],

(9) R)SUT @XV Vprobabilities _a`b@ @WV(or finite

(10) R)SUT measures), that is probabilities in the idempotent semifield ZY\[^] R)SUT , can be introduced. In Polish spaces, they necessarily have a l.s.c. density [1] and the “measure of a set” cor@XV probability theory responds to the minimum of a function (the density) on this set. Then, -RIST leads to a probabilistic formalism for optimization and optimal control, that we call decision theory. This has been developed in [9, 12, 4, 2, 5]. Random variables (called decision variables) correspond to changes of variables or constraints parameters on an optimization problem. Markov chains (called Bellman chains) correspond to optimal control problems. Weak convergence of decision variables corresponds to the convergence of value functions. It is related with the epiconvergence introduced in convex analysis [6, 16] and is equivalent to the weak convergence of capacities introduced in [22, 21]. Classical limit theorems of probability (law of large numbers, central limit theorems) correspond to asymptotic results for the value function of optimal control problems. Since the images of the Cramer transform are l.s.c. functions, we may say@WV that the Cramer transform sends classical probabilities (resp. finite positive measures) into -RISUT probabilities (resp. @WV edgf finite -R)SUT measures). For instance, if is the Dirac measure at point c , (where df 43 V_ for 3i h c and df Fc kj ), which is the density of the

(11) R)SUT @WV Dirac measure. #v f If t @Wno n kp @Wnoq is the normal law lm4c of mean c and standard deviation , 4c 3)!s t r u wyx which has the same stability property as the Normal law. Moreover, the Cramer transform leads to another correspondence between probability theory and decision theory. Indeed, it sends the convo@XV lution of two probabilities into the SUTAz -convolution of their images; but the -R)SUT equivalent of convolution is the inf-convolution of densities. Similarly, the Cramer transform sends independent random variables into independent decision variables. In this paper we study the continuity of the Cramer transform when both classical probabilities @XV and

(12) R)SUT probabilities spaces are endowed with the topology of the weak convergence. Logconcave measures and functions appear naturally in probability, heat equation theory, optimization [23, 19, 20, 10]. Typically, normal laws and Wiener measures are logconcave. We show here that the. RR n ˚ 2841.

(13) 4. M. Akian. Laplace transform has almost the same behavior as the Fenchel transform where convexity of functions is replaced by logconcavity of measures. We then prove the equivalence between vague and weak convergences on the subspace of logconcave finite measures over a finite dimensional vector space and the bicontinuity of the Cramer transform on this subspace. Counter examples show that the logconcavity condition cannot be relaxed. For the infinite dimensional case, we prove a similar continuity result under assumptions on similar to that of [8].. 1 Notations and definitions @WV. @XV. Let us first recall some definitions and results concerning -RISUT V_a`b@ @WV probabilities. The term -R)SUT algebra refers to the idempotent semifield ZY\[^] denoted Y . The neutral elements ) R U S T V V_ j d for the R)SUT and laws are respectively and and are denoted and . Let us note that the order associated with the idempotent RISUT law is the reverse order of Y (the order associated with an ). Therefore, inequalities and limits hold in idempotent law is defined by the reverse order compared to that usedVfor The name _k` general idempotent1 or classical v probabilities. # byv the order relation Y will also be given to the set Y [] endowed with the topology defined ( @ ( C which is equivalent to that defined by the exponential distance 43 . Let us finally Y note that, since is an upper bound of Y (for the classical order), bounded sets of Y correspond to lower bounded sets.. .

(14). . . . . . . . . . . . . . Definition 1. Let be a topological space and the set of its open sets. A finite -RIST @ measure or cost measure on is an application from to Y such that. (b) [ H H (a). . V_. US TAH z @WV. . H. . for any. . H. . . . . . . @XV. . idempotent. .. j and it is null if V_ . %#" @XV If 5 is a bounded function from to Y , SUTAz! 5'%$ is a -RI@ ST idempotent measure. If has this form, 5 is called a density of . Any cost measure on admits a “maximal” ( of : (in terms of order) extension '& to the power set ) & *) ",+.7- 9 . :" %0/ / If is a separable metric space, then has necessarily a density. Its “maximal” density #%#5 is equal ` to 51& F3 2& ] 3 and is l.s.c. [1, 17]. For general topological spaces, 3P& %4 SUTAz 51& 43 for any compact set 4 [1]. Then5 has a density if and only if C67& is regular or is a capacity in the sense of [22], that is - STgd z - compact 8:9 & %4 . @WV In the sequel, denote respectively the classical functions - ; and - andj -R)SUT characteristic V_ d of the set ) - : ; 43 if 3 and 0 if 3# h ) , if 3 and if 3 h ) . If ; ) 4 3 ) # ` d d ) ] 3 , ; and are denoted by @ ; and . Given any cost measure on , the Maslov integral with respect to is the unique Y< – linear form, denoted also , on the set of upper semicontinuous (u.s.c.) functions = ! Y< It is a. -RIST. probability if in addition. INRIA.

(15) . On the continuity of the Cramer transform. %=. . = &. . =. =. 5. d- . . . ) %=. . ). such that H and [18, 1]. If the cost measure when H for % V has a density and 5 is its “maximal” density, then U S A T z # # 5 . Therefore, @XV d # the equivalent to the Dirac measure in point 3 is the cost measure with density .

(16) R)SUT Using this formalism, the weak convergence of cost measures is defined as usual :. . 9 = $. . Definition 2. We say that H weakly converges towards , denoted for any bounded continuous function k! Y .. = . . . 1& %$. . H !w. H = ! H %= . . , if. Using the correspondence between cost measures and their densities, we will also speak of the weak convergence of functions. Theorem 3 ([5, Th. 5.2]). Let iff . . H. and. . SURkH SUTAz H

(17) SURkH 79 : Hq . . @ . . be cost measures on a metric space. . . Then. . H !w. . for all closed for all. . . . . . . Let be a locally compact topological space and suppose that H and have densities (this is the case if is a finite dimensional vector space). Then, they can be considered as capacities in the sense of [22] and the vague convergence may be defined either by the condition Hq ! H for any continuous function with compact support or by the following definition which coincides with that of [22].. . %=. =. . Definition 4. We say that. . H. . vaguely converges towards , denoted. . . USURkH SUTAz H %4 %4

(18) USURkH 729;: H . . for all compact for all. . . . . 4. H !v. . %=. if. The vague convergence is related to the epigraph convergence of functions defined in convex analysis [6, 16]. Definition 5. Let. 5H. and. 5. be l.s.c. functions. We say that. epi-converges) towards 5 , denoted 5H. :$. @. (a) (b) . $. @ . $. H !H $;H ! H. ! 5 , if USRaH STgz 5WH %$;H k 5'%$ , @

(19) U SRaH 729 : 5 H %$;H 5 %$ .. $. $. @. H. and 5 , then. 5H. converges in the epigraph sense (or. . Proposition 6 ([5, Th. 5.7]). If densities 5. epi. . H !. v. . . is a first countable topological space and iff 5. H ! 5 epi. H. and. .. The tightness is also defined in a similar way to classical probabilities (recall that Definition 7. A set of cost measures is tight iff. 5. 729 :. compact. RR n ˚ 2841. % STgz. 4 . V_ . . . have l.s.c.. V_. )..

(20) 6. M. Akian. The vague and weak convergences are equivalent for tight sequences and any tight sequence of cost measures in a metric space admits a weakly convergent subsequence [5]. Let us also note that any sequence of l.s.c. functions on a separable metric space admits an epi-converging subsequence [6]. We also use the following V_ result proved in convex analysis [6, 16]. A l.s.c. function 5 ! Y is proper iff 5h .. < . . Theorem 8. Let be a finite dimensional vector space. The Fenchel transform is bicontinuous on the set F of proper l.s.c. convex functions endowed with the topology of the epi-convergence.. . In a reflexive Banach space, the Fenchel transform is bicontinuous on F endowed with the topology of the Mosco-epiconvergence defined as the epi-convergence (definition 5) except that the convergence of H towards holds in the weak sense in (a) and in the strong sense in (b). The Moscoepiconvergence is also equivalent to the vague convergence of cost measures defined as in definition 4 except that may be any bounded closed convex set.. $. $. 4. 2 Statement of the results and counter examples . For any locally convex, Hausdorff topological vector space (l.c.h.t.v.s.) , qF will denote the set of non identically null u.s.c. logconcave functions, that is the set of functions ! Y L such that C U F . If is a closed convex subset of a l.c.h.t.v.s. G , any u.s.c. logconcave j outsidefunction on can be extended to an u.s.c. logconcave function on by taking G 4 3 . Then @ ` F ] -G 43 j 3 . Generalizing the definition of [24] to the infinite dimensional case, we obtain. =. =. 0=. =. =. =. . Definition 9. A positive measure defined on the Borel sets of a closed convex subset of a l.c.h.t.v.s. G is said to be logarithmically concave (or logconcave) if v. ) @ . for any pair measures over . !). V. of convex subsets of p will be denoted by . . ) r (1)

(21) j @ and any ; . The set of finite non null logconcave . . !;C. 4 . p @ p 4 ] -G F mj ` Again, . . v Remark 10. From the definition of logconcave measures,we see that if is a continuous linear apv v ), the positive measure plication between the l.c.h.t.v.s. and (in particular if r defined by for any Borelv set of v is logconcave as soon asv is logconcave. Inr r v if and arev convex sets of , r and r are convex and r V C deed, ;V C r r . In particular, marginal laws of logconcave measures are also logconcave measures.. . . .. ) *) ; *). *). ). . ). . . . !). !; . . By theorems 1 and 2 of [24], any non null positive measure on a finite dimensional vector space which admits a density =4 with respect to the Lebesgue measure of is logconcave. Moreover, as for logconcave measures, marginal laws of measures with logconcave density have logconcave density [24]. Indeed, logconcave measures have essentially a density. This is stated in the following result proved in appendix.. . =. INRIA.

(22) On the continuity of the Cramer transform. 7. . Proposition 11. A finite non null positive measure on the finite dimensional vector space is logconcave if and only if there exists an affine subspace of containing the support of and such that has a density = with respect to the Lebesgue measure over (if the dimension of is zero, the Lebesgue measure is reduced to the Dirac measure and is constant).. . . =. =. Theorem 12. Let be a finite dimensional vector space. The Cramer transform is injective and bip F to its image in F endowed respectively with the classical weak convercontinuous from @XV p 4 vague gence and the -R)SUT weak convergence topologies. Moreover, for sequences in convergence and weak convergence are equivalent. For the infinite dimensional case, we adopt the minimal assumptions on stated by Bahadur and Zabell [8, 7, 13] for the construction of the Cramer transform and the Cramer theorem. Theorem 13. Let G be a l.c.h.t.v.s. and be a closed convex subset of G such that is a Polish p 4 to F endowed space for the induced topology. The Cramer transform is continuous from @WV respectively with the classical weak convergence and the -R)SUT weak convergence topologies.. . . Remark 14. In [7] it is proved, using the Cramer theorem, that if A9 : : closed convex, then R = . This can> also be done directly using Hahn-Banach theorem. Indeed, if i 3 h , there A@ j and > @ B j for any ^ . Then $ for any exists G such that P 3 B > @ V_ j and F3 m729;: 3 B . This implies that if is a closed convex subset of G the image of p 4 is included in F .. . . .

(23) . . . . . ;. . v v # 1 Example 15. Let us first give a counter example for the case ( of general " ( positive measures or proba Q Q 3 where bilities. Let H be the probability on Y with density PH. . V _ 3 5WH 43 . "! . . . 3. for for. 3. j. . . j . . %$. d. . @XV. Let us first note that 5 H is a sequence of l.s.c. functions for the -R)SUT weak d . 8d v . such that 5 H !. convergence, but that F5 H and then does not epi-converge towards L d $ d . Thus 5 H gives a counter example to the continuity of the Fenchel transform for general (not necessarily convex) proper l.s.c. functions. w We have also H ! for the classical weak convergence, since. . . ++ &. and. , = @+O. ++ =. 43. . &. 1. =. 43. . 1. #. # " L # = u x H 43 " L # "H ( ' v*(. ++. + , @ ; N V.0/ Ho43 C =N j + %=. where denotes the continuity modulus of However . ' 21 H r. RR n ˚ 2841. #. . 4t 3. =. . j. w. (. v)( # 1 3 #1 3. = /#. v)( # 1 ( LH # )v ( # 1 3 @ " L # ( 3 ". in .. V_ for j @ j ! H r j ; for.

(24) 8. M. Akian. d ! .. Then similarly to. . H !. epi. L#. 4 5 H we have U D H ! H d v h = d .. . %$. #. pointwise and in the epigraph sense. Then. This example shows in addition the analogy of behavior between the Fenchel transform and the logarithm of the Laplace transform.. .. Example 16. Let us give another counter example concerning stable distributions. Let us consider j j ` so the a stable distribution with order on Y . Then, either the domain of is ] Cramer transform of has no interest. Either it is equal to one half space, for instance Y L and then ' j and ' V_ for j . Considering now the proba U D for O ' ' , we get oH ! w . However U = H U = u , x ! H d . for bility H H. . . . and H. . . . . d v. . %$. . . +. . d. . . . . !. . L#. . . h any . This shows that the law of large numbers for i.i.d. ! random variables with a stable distribution of order eh be transferred from probability @WVcannot to optimization by the Cramer transform. Indeed, the

(25) R)SUT law of large numbers does not work < identically costed decision variables with non tight cost density such as in general for independent @ j j and r V r < ). (with ; ;+R C 3. #. epi. . . . . . .. ;. @WV. law of Remark 17. In [2], we noticed that if the Cramer transform were continuous, the -R)SUT @WV large numbers [25, 4, 2, 5, 12] and the -R)SUT central limit theorem [25, 4, 5] may have been consequences of the classical ones. Example 15 and 16 show that this is not the case. Indeed, the laws appearing in these results are in general not the images of probabilities by the Cramer transform. By its definition a proper l.s.c. convex function. Moreover, if for instance Y , - is necessarily #v f differentiable in the interior of its domain. is analytical and then infinitely n p Fc @Xno^ 3k! r w j appearing Then, the l.s.c. function with , h and < in central limit theorem [4, 5]< is not the image of by the Cramer transform. Indeed, p 4c @Wno2 ' c V r no with r V r < a probability , then it is not regular in 0 except for integer p @Wno values of ; . But in that last case and if = ,

(26) G Fc c and the variance n mj J of G is zero, then G kc and .. . . . . + + . . ++ . ;. ; . . . . p. .. Plan of the proof of Theorems 12 and 13. Let 4 p denotes the set of finite positive measures on p @ ` the Borel sets of . If is a closed subset of G , 4 ] -G F j . If is a Polish space, the topology of the classical weak convergence is metrizable. Then the continuity p F to F both endowed with the weak convergence of the Cramer transform from a subset of w topologies is equivalent to the sequential continuity that is H ! = H ! w = . Since @WV @WV the topology of the

(27) R)SUT weak convergence is also metrizable in the subspace of tight -RISUT p measures [3], the bicontinuity of the Cramer transform p on F is equivalent to the sequential bicontinuity, if = is proved to be tight for any @WV F which is done in section 4 and 5. Since, in a reflexive Banach space, the

(28) R)SUT weak convergence implies the Mosco-epiconvergence and the Fenchel transform is bicontinuous on l.s.c. proper convex functions for the Moscoepiconvergence, the continuity of U is required at least in the following sense :. . . . .

(29) . . . . . H !w. y U . H !epi . (2). INRIA.

(30) On the continuity of the Cramer transform. 9. # H ! H j , Q !w $ U j but. However the continuity of U for the weak convergence is not required. Indeed if 3. . # d#. +. d. . # e d # I = Q !. . and U D = , = # Q Q ! weakly j Q U D Q does not converge towards . w. 3 H !epi. If (2) holds in finite dimension, the continuity of the Cramer transform will be a consequence of v apply Theov the tightness of H for tight sequences H [5]. In the infinite dimensional case, we 2 2 w p w v v v v H r rem 12 to marginal laws. Then H ! = H r !8 with and in 4 implies in =ZY (since H G . If H is tight, there r !w r and r r ) for any w , where is a l.s.c. function on . By exists a converging subsequence also denoted H ! v v the convexity of v H , is convex and can be extended to a l.s.c. convex function on G . Then, 2 H r ! w r = r for any G and = . By unicity of the limit, the sequence = H weakly converges towards . Again the continuity of the Cramer transform is a consequence of the tightness of H for tight sequences H . We then reduce the proof of Theorems 12 and 13 to three parts : the continuity of U = for the epiconvergence at least in finite dimension (section 3), the tightness of = H in finite dimension v (section 5). The tightness of for any p F , (section 4) and then in infinite dimension the injectivity and the continuity of r in finite dimension are proved in section 4.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Continuity of. We use the following lemma which is the analogue of a result proved in [2] for the Fenchel transform. @WV The vague and weak convergences are denoted identically for classical and -RISUT measures.. Lemma 18. > Let and G be as in Theorem 13 and G A@ p G G ! 3PB is continuous. Let H and. . USRaH STgz oH H. . and. R. =NF3. 4. . Proof. Let.

(31) . Since is a Polish space, set , we get (3) when H. RR n ˚ 2841. . ' ' A@ @ ' V_ oH ! H . 2 ; USRaH 729 : oH 1 # 5 # 0 be a compact uniformly equivalent to the topology of . @ j subset of( *- ,/. , a distance !;C =NF3 is bounded and continuous. Since H ! w and . Then, & ( *

(32) ,. #'0 1 /@ ' H =NF3 H 43. we have. 4. be endowed with a topology such that F3 F . Then H ! w implies ' H !H in G . . <. ' USURkH SUTAz H . &. 5. ( *-,/.. # 01. F3. @X'D. (3). and. . is regular (or is a capacity), then taking the supremum over the compact . For the case H !EH we obtain. USURkH SUTAz H H. . (. v. Q. 5. & H. 5. ( *-,/.. #'0 1. F3. .

(33) 10. M. Akian. where. @ . . %4. . A@ # 729;5 : > H C 3PB . #729 %#:5 > H C A @ P3 B V # 27 9 : .. .

(34). j. A@. H C

(35) >. . C 3PB. . Since the first term tends to and the second is small when is large and is small, we obtain (3). For the second point, we use that H ! H not only for bounded continuous functions but also for functions continuous and equi-integrable with respect to H , that is such that. =. %=. &. . . uniformly in . Since USRa729. ' :H H . &. *-,/.. V_. . # 0

(36) ( *-,/.. . %=. 1 # N= F3 oH 43 !. , we have . #'0 1. . H F3. . #. j. H ' 4 for v ' ( r H . large enough and. . . =. . . !. #. j. uniformly in . Remark 19. In finite dimension, the proof of inequality (3) only requires the vague convergence of H towards that is H ! H for any continuous function with compact support.. . . %=. %=. Let us first give a proof of the continuity of the Fenchel transform that can be adapted to prove for F5 . the continuity of U . If the cost measure has a l.s.c. density 5 , we write . . H be a sequence of cost measures on H !epi in G . . By a result equivalent to Lemma 18 for instead of [2], we have for any sequence H ! H. Proposition 20. Let , G. and G. with convex l.s.c. densities, then Proof.. . . . be as in Lemma 18. Let. H !w. . ' j. j since the constant that is condition (a) of definition 5 is fulfilled. Moreover H ! H j function is bounded. I v that v SURk79 : H H ' R for some ; v and let prove that G such Let v v . We denote by 5 and 5 H the l.s.c. 2 densities of r and H r ( r is the cost measure on Y such that r ) for any open set ) of Y ). The convexity of the density of r * ) H implies that of 5 H . Then #729;%: X3CE5'F3 V_. . USURkH SUTAz H H. implies v that 3. H. . r !w. . . C\ v 5 F3 decreases in some point that is 3 r CE5'43 r 3 t CE5'F3 t for 3 r m3 t . Since r , 5 H !epi 5 and there exists 3 H ! H 3 r such that 3 H Ck5 H 43 H 3 t Ck5 H F3 t and. INRIA.

(37) On the continuity of the Cramer transform. 3 H k 3t. . 11. for large enough. By the convexity of 5 H , this implies that 3C. and. H ' 729 # : 3C 5 H F3 # 27 9 # : 3C 5 H F3. '. 5 H F3 . . decreases for 3. . 3 t. ' #P# : H 729 : 3C 5'F3 ' r L ' ' H for such that R with 2v ; . Using in ^ ; , we can construct a sequence H ! H ; such that ^ R and then for any G which proves for any. Then. USURkH 79 This proves that USURk729;: H. the left continuity of ! ' . USURk729;: H H H property (b) of definition 5.. Let us now give the proof of the continuity of U which is a direct adaptation of the previous proof. Proposition 21. Let , G. and G. . be as in Lemma 18 and let. y U oH . . H. and. . . p 4 . Then. !epi U y in G ' v v Proof. Again, by Lemma v ' 18, US Ra'Sv Tgz4H U v o H H U for any H ! H . Let us prove that USURk729 : H H for any R . Since H r and r are logconcave, ' # can suppose that Y . In that case, H is a r ; and oH r ! w r ( , , we # # Dirac measure ( , or has logconcave density, then oH are not necessarily finite ( , logconcave measures such that converges towards the finite logconcave measure H vaguely . We are then reduv ced to prove that H ! with H not necessarily finite logconcave Radon measures and a finite UY . logconcave measure over Y imply USURk79 : H oHoUY V_ j @ ; 2 j . Then, since j Since ZY , we can suppose after translation that % o @ V ; V_ @ UY . oH. . . . . o@ V ; 2. ! . !w. decreases in some point. @ V. j.

(38) . v ; , that is ; 2 ( C ; @ 2. . (4). with . o@ PV If has a density then the sets are -continuous for any . Otherwise, from the logconca j @ j @ vity of , is a Dirac measure at a point 3 . After translation, we can suppose that 3 @ V v and then the sets are again -continuous for any integer . Therefore, if =H ! , (4) is valid for oH instead of and large enough. Since oH is logconcave,. . . . RR n ˚ 2841. . ;. . . ;. ;. H @ V ; t H C ; @ 2 H V ; @ V .2. . . ;.

(39) 12. M. Akian. V ; @ V . and applying (4) we obtain H ;, H V @ V V ; and. oH j @WV_. . oH j @ . . Increasing. . . 2PV. . @. . V. V . ; . . ; . @. . V . . . . Taking the limit when. ;C. . ; ( v C V ; . @. . . V. ; . j. and. 2 (5). . _ @j C . Then, v UY( . and and decreasing , we obtain an analogue inequality for =H. USRaH 729 : o HNUY. . and so on. Then for. . % o H . . ; ( v o Ho j ;C. H @ V ; 2 v ( H @ V. v (. V. ; ;C. . goes to infinity, we obtain the expected inequality.. . . Y , the vague convergence of H From the previous proof and Remark 19, we see that if towards is only required. This fact is also true in finite dimension and indeed allows to prove the equivalence between weak and vague convergences.. . . Proposition 22. Let. H. . . and.

(40). p UY , then. oH.

(41) oH. !v. !w. .

(42). Proof. The vague convergence of general finite measures on Y. . U SURkH 729 : H %4 USURkH SUTAz H .

(43) . %4. . . . is equivalent to the conditions. for all compact for all open. . . 4.

(44). and the weak convergence is equivalent to the same conditions but with closed sets in place of com

(45) USRa729 : H oH ZY C for pact sets. Since the inequality on open sets implies SURk79 : H oH . any closed sets, we can pass from vague to weak convergence only by proving USURk729 : H oH UY 1 ZY or equivalently USR H oHoUY ZY . This property has been proved in dimension 1 1 true in that case. Let us prove in the proof of previous proposition and then Proposition 22 is v H and this property by induction on . We suppose that it is true for C and consider p UY vsuch that H ! v . Let be a bounded convex set of Y such v that UY r j j H and UY and consider the following measures on Y r : H r v v and . These measures are logconcave and finite, is non null and since j v non null for large enough. Moreover H ! v . , H is USURkSUTAz H H ZY r UY v r SUT By v induction, we get H UY for any bounded convex set such that ! H ZY r r is -continuous with positive measure. The positivity condition can be eliminated by Y r v v adding a disjoint convex set satisfying the previous conditions. Considering now the measures H UY r and o on Y and noting that the proof of ProposioH U Y r tion 21 only used the convergence of NH towards for -continuous bounded convex sets ! H ZY . , we conclude that H ZY. .

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(62) ) 4 4 5 5 4. . . . INRIA.

(63) On the continuity of the Cramer transform. 13. . Tightness of and bicontinuity of the Cramer transform in finite dimension. 4. We use the following lemmas.. //. Lemma 23. If has finite dimension and denotes a norm on , the following propositions are equivalent for any finite positive measure on (a). R. j SUT. (b) there exists . j SUT. R. . . . . j. such that v. r /+" / . # 1 43 . " (. V_. . . . # 0 1 # 1 ' " ( *-,/. " ( V_ j@ Proof. , 43 43 . Then R (b) (a) if v . j@ V_ (a) (c) If for any such that , . R , then r @$+$P@X $ @ $ + P X @ @ $ @ + $ P@ ` 5 and ] # #0 C C %

(64) ( . Then (c) (b). Let be a base of . %

(65) > ( *-,/. A @ / / / /. . r r r 3 ' 79 : , , 3PB defines a norm equivalent to . Therefore 3 4 3 , and & ( # 1 F3 , %

(66) % 4 ' v j SUT R r for any , there exists j such that the right hand Since is finite and (c). . for any.

(67). .

(68) . side of the previous inequality is finite.. .

(69) . p p F , R is open and then # 0 and are injective in F . ( *-,/. p ^ p v Proof. Since and implies F . E R F p have to prove j SUT RE for any p 4 . Using lemma 23 and the fact that , wer only UY for any to the one dimensional case. we arep reduced v # 1 then RE Y is open. Otherwise has a density v ( Let ZY . If is a Dirac ( " measure V_ with 5 convex. From ZY 3 , we see that 5 strictly increases somewhere. j . Then, 5 ' 5 F 3 C 3 By the convexity of , this implies that increases for 3 large enough # #% and V_ V_ j 79 : ' 3 CE5'43 and by symmetry we obtain, for some , 79 : v 3 C 5'43 ( ' V_ which by lemma and 23 allows to conclude. j implies . R The openness of that is characterized by its Laplace transform . Since U is a l.s.c. convex function, it is also characterized by its Fenchel transform . Then and are injective on p F . #'0 =1 H when H ! w . From the previous Proof of theorem 12. Let us now prove the tightness of " ( * / , . V_ /+" / . j results we know that there exists @ such that / / F3 %

(70) > Afor any such that @ j such that 3 r 729 : , 3PB . Since (by the proof of Let us choose a finite set Lemma 24. For any. . . . RR n ˚ 2841.

(71) 14. M. Akian. proposition 21) USURk79 and . Then,. ' ' : H H H F 3 . . . 4 . ' , we have H . for. . < > A@ P3 BNC U H ' > A@ / / ' 3PBNC 4 3 C 4. %'& , 729;: %

(72) ,729 :. . . . for any. This shows that the sequence = H for is tight and by the previous section and the plan of section 2 we obtain the weak convergence of H towards . v 24, the Cramer transform is injective and we can consider the continuity of the inBy Lemma p F . We only have to prove that for any H and p 4 , verse map r on the image of w w H !y= implies H ! . But since any sequence of finite measures on Y admits a vaguely v for a subsequence of H also denoted H . This implies converging subsequence, we have H ! w the logconcavity of and then this is equivalent to the weak convergence and implies H !y= . and by the injectivity of , we get Then = . By the unicity of the limit we obtain w H ! .. . . . . . 5. . . . . . . . . .

(73) . . . . The infinite dimensional case. From Theorem 12 (proved in sections 3 and 4) and the plan given in section 2, the proof of Theorem 13 @ p w F . Since oH ! w and oH is reduced to the proof of the tightness of NH when H ! and is a Polish space, the set composed by the measures NH and is tight. Moreover there exists V_. j 4 such that for any measure in this set. We say that a set of positive measures is bounded when it satisfies this last condition. The following result concludes the proof of Theorem 13.. . . . . . . . . . . . . . . Proposition 25. Let and G p 4 and bounded subset of Proof. Let that. . be as in Theorem 13. The image by the Cramer transform of a tight is a tight subset of F . p 4 . There exists a compact subset 4 of such be a tight and bounded subset of . 4 . ; 4 . . for any. . . . (6). 4. C 3 ) hull of . Since is Let us consider the closed convex and symmetric (stable by 3 ! convex and complete, the hull' of is still compact and satisfies (6). We it again by . Let 79 : #% > A@ 3PB . From the symmetry ofdenote be its support function, q , is a seminorm on @ ' ` G d and its unit ball. is a closed convex set, > A@ Since we denote by , % ] < > A@ G q ' . ^ F3 F3 79 : 3PBC . Then 3 iff 3PB for any and the. 4. . 5. 4. . 5. J def . #5. #%.

(74)

(75) . gauge function of is F3 SU TAz . Let us now upper bound for any ' and . Let us fix. . . . 5. ;. . %. <>4. A@ 3P B . . Forv any . @ G^ , . Then r ' C ; ; 4. . 729;: ,. 4. 4. . ;. . and. 4. v. . r is logconcave @ r on Y N C7; ; NUY .. INRIA.

(76) On the continuity of the Cramer transform. 15. However, by the logconcavity of , we have for. @ . . N2!;. N C7;. then. . ' . (. 5. . @ r K ; Q C7; V. @ ; V. ; .

(77) . ,. . ;. V-. @ V. N2 C7;.

(78) . (t. Therefore if. v. & (

(79) # . . 1. . A. . 4. . ;. H. . . 3.

(80) N2 C _ @ ; V # (

(81) r L t H C7; V. @ ; V . H r (

(82) / F ' ;C. 5 . ' . Then, for some constant independent of , If is an upper bound of F %

(83) for obtain. #% , we. 43 C U C . Then SUTAz SUTA z = 43 sets implies the tightness of .. NUY. . QK. . . . ' DV % < > A@ U F ' ; . implies 27 9 : , J 3PBC 43 C U C ; which by the compacity of. . . . Proof of Proposition 11. . . As was said in section 2, one implication is a consequence of theorems 1 and 2 of [24]. Indeed, if the support of , A9 :;: and is an affine space, the logconcavity of is equivalent to that of and if has zero dimension, is obviously logconcave. @ V j @ the open ball of Let be a logconcave measure on . If we denote by 43 @ $V 3 ;@ V @ center 3 and radius for the euclidian norm, we have C F3 j 2 C @ 3 j . I Then,;the of is convex. Indeed, if 3 and A9;: : V , for@ any @ 2 support j , F3 j @ and j and 3 by the logconcavity of , 2 C for any . Let be the affine hull of A9 : : , is logconcave and we may then suppose without restriction that . Then, by the convexity of A9 : : , its interior is non empty [26]. If has zero dimension, is a Dirac measure and we have nothing to prove. C _m@ j is convex, NF3 V 3 Let us first consider the one dimensional case. Since _ @ ` ] 3 Y @ NF3 a C 3 is a logconcave function on Y and R def h j is the classical domain of the convex function C and it contains the interior of A9 : : . Indeed, if 3 U S T A9 : : , V @ 2 j . Moreover, since 3iC A9 : : for small enough, then 3 V j @XV_ 2F3^C 3 is a nondecreasing function R . By the convexity of C U , is ST g9 : : V j @WV _ continuous in the interior of its domain, then in SUT A9 : : @WV_ . By symmetry, we obtain that is continuous on Y . Indeed, 43 2F3 UY C NF3 is nonincreasing, logconcave. . . . . . . . . . . . = =. RR n ˚ 2841. . ,=. . ; ; . . . = . 4. . ). =. 4. =.

(84) . 4 . =. . ; ; ;. =. .. . %4. 4 = =.

(85) 16. M. Akian. 4 . . . 4 V =. _ @j. and its domain contains SUT A9 : : and then R ST A9 :;: ` aj mj V implies C SUT A9 :;.: This j @WV_ .that for any and also R ] Y and then A9 : : Since C finite right and left derivatives in each point of its open doV is convex, j @WV_ itandadmits main ST A9 :;: so does . Since the complementary of this domain is stable j there, the left derivative of exists and is null. Then, v 43 by # v and negative translation . v @ L USUR. Y . Moreover since C j is convex, is exists and is finite for any 3 again a logconcave function and then it is continuous in the interior of its domain j and if in addition Pv F3 which j , wecontains SUT A9 :;v : . Indeed, U S T A9 : : , then 43 if 3 obtain

(86) F3 kj . Since UV j @WisVa_ concave N NF3 for and nondecreasing function, this implies mj any which leads to a contradiction with 3 3 and then F3 SUT A9 : : . v exists, We have then proved that is finite and is logconcave on Y and that it is strictly positive and continuous and then equal to P in SUT A9 : : . Moreover is null on A9;: : and mj . This implies that is a density of on Y . -SUT A9 : : We now solve the dimensional case by induction. Let us first note that the previous proof is still valid if is not a measure on Y but only a bounded, positive, logconvave (1)), (satisfying AV additive and if nondecreasing functional on convex sets (that is such that [ if ) with a support and (the support is defined as for @ 2 with j fornonempty j ).interior A9 : : iff 43 measures, that is 3 all Indeed the logconcavity implies the continuity of without assuming the continuity of and the continuity of implies the continuity @ of on convex sets (intervals). For and is neither empty, we have instance if H H j H 1 H and H and then H N H C H . The conclusion of the proof "is at least that has a density which is logconcave and finite everywhere in the sense that F3 3 for any interval . Let us consider a bounded, positive, logconcave, additive and nondecreasing functional on the H convex setsv of Y , with a support with nonempty interior, and suppose that any functional of this H 1 and logconcave everywhere in the sense that type on Y r has a density which is finite " $+ v (where are intervals of Y ) and that F3 3 for any rectangular boxes H. . 4. . . ). . 4 . =. . . . . =. 4. . . . =. 4. =. %4. . . =. =. 4. 4. . . 4. 4. . =. % . = . =. = . . . 4 4. . 4. =. *. =. . . =. . 4 4. ). =. =. 4. . %$ = . . = . . . ). . ,=. . . 5. 4. r. r. 4. @j C $H v @ j C . r $ + v r USUR U SR H Q AK v K r r H r the functional 5 !

(87) 4 is a bounded, positive, logFor any convex set 4 of Y 5 1 concave, additive 5 and nondecreasing functional on the " convex @ sets of Y . If A9;: : has a nonempty 4 F3 r 4 3 r for any interval where interior, then has a density, that is

(88) @ F3 r V C r @ j 4 F3 r 4 U SR r 5 j K 5 is finite for any 3 this is again true. If now A9 : : r Y . If ` ` is non empty with empty interior, it is then equal to some singleton ] . Therefore, g9 : : m0] 4 [ Y

(89) SUT .4 . The convexity of A9;: : and the5 fact that it has non empty interior implies that A9 : : Y -SUT .4 5 < ` 4 , either and then 0] set such that SU 4 ] ` j . If now 4 ` is a convex T .4 5< has a support with empty interior and then ] 4 UY iST 4 j , either has a F3 . . +$. . . 43. V. . INRIA.

(90) On the continuity of the Cramer transform. 5. 17. < `' j ` density and then ] . The two cases contradict the previous conclusion, ] then all convex sets are such that g9 : : is empty or has nonempty interior and then such that @ v exists and is a# density of . F3 r @ H v @ For any 3 is a bounded (by 43 Y ! F3 r r H which is finite),# positive, r r Y ,# K 1 1 logconcave, additive and nondecreasing functional on the convex sets of Y @ " @ t @+$+P@ r . If A9t +: $: K has non F3 r 3 3 H 3 3 H for any empty interior then K has a density, that is F3 r rectangular boxes , with. 4. 4. 4. #. . . . 4. . 4. 5. . 5. 4. 4. 43 . Q USUR. . $$. K. F3 USUR . K . is null,v this is also true. If now A9 v : : H of a C affine subspace oft Y @ j ` j ` dimensional H t H If. #. vK. V. . 5. . @ C r j + $ r H. @ C H j . is nonempty with empty interior, it is then a subset. v that r . By rotation andV translation, @ j wej can ` suppose j H ] j Y t and then 243 r C ] Y t j . Then 43 ] r v )Y for small enough. Exchanging 3 and 3 coordinates in the previous reasoning, we see that # cannot be strictly positive on 4 ! F3 r V C @ j 4 Y H t hasr necessarily# a density and then @$ a singleton which contradicts the assumption on K . Therefore, K has always F3 r as density. . . . and has as density (on rectangular boxes). The induction is then proved and if is a measure, is also a density of on Borel sets. Moreover, since is continuous in the interior and exterior of kj , we can repace by its u.s.c. envelope. A9;: : and A9;: :. . . . . References [1] M. Akian, Densities of idempotent measures and large deviations, Rapport de Recherche 2534, INRIA, 1995. [2]. , Theory of cost measures : convergence of decision variables, Rapport de Recherche 2611, INRIA, 1995.. [3]. , From probabilities to cost measures via large deviations, Rapport de recherche, INRIA, 1996, to appear.. [4] M. Akian, J. P. Quadrat, and M. Viot, Bellman processes, 11th International Conference on Analysis and Optimization of Systems : Discrete Event Systems, Lecture notes in Control and Information Sciences, vol. 199, Springer Verlag, 1994. [5]. , Duality between probability and optimization, Idempotency (J. Gunawardena, ed.), Cambridge University Press, 1996, to appear.. [6] H. Attouch, Variational convergence for functions and operators, Pitman, 1984. [7] R. Azencott, Y. Guivarc’h, and R. F. Gundy, Ecole d’été de probabilités de Saint Flour VIII1978, Lecture Notes in Mathematics, vol. 774, Springer Verlag, Berlin, 1980. [8] R. R. Bahadur and S. L. Zabell, Large deviations of the sample mean in general vector spaces, Annals of Probab. 7 (1979), 587–621.. RR n ˚ 2841.

(91) 18. M. Akian. [9] F. Bellalouna, Un point de vue linéaire sur la programmation dynamique. détection de ruptures dans le cadre des problèmes de fiabilité, Thèse, Université Paris-IX Dauphine, Paris, 1992. [10] H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Prépoka-Leinder theorems, including inequalities for log concave functions, and with an application to the diffusion equation, Journal of functional analysis 22 (1976), 366–389. [11] H. Cramer, Sur un nouveau théorème limite de la théorie des probabilités, Actualités Sci. Indust. 736 (1938), 5–23. [12] P. Del Moral, Résolution particulaire des problèmes d’estimation et d’optimisation nonlinéaires, Thèse, Université Paul Sabatier, Toulouse, 1994. [13] J. D. Deuschel and D. W. Stroock, Large deviations, Pure and applied Mathematics, vol. 137, Academic Press, London, 1989. [14] R. S. Ellis, Entropy, large deviations, and statistical mechanics, Springer Verlag, New York, 1985. [15] M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, Springer Verlag, Berlin, 1984. [16] J. L. Joly, Une famille de topologies sur l’ensemble des fonctions convexes pour lesquelles la polarité est bicontinue, J. Math. pures et appl. 52 (1973), 421–441. [17] V. N. Kolokoltsov and V. P. Maslov, The general form of the endomorphisms in the space of continuous functions with values in a numerical commutative semiring (with the operation R ), Soviet Math. Dokl. 36 (1988), no. 1, 55–59.. . . ´ [18] V. Maslov, Méthodes opératorielles, Editions MIR, Moscou, 1987. [19] R. Michel, Logconcavité, e´ quation de la chaleur et viscosité, International journal of Mathematics 3 (1992), no. 4, 499–524. [20] R. Michel and M. Volle, Logconcavité et inégalités intégrales, Revue Roumaine de Mathématiques Pures et Appliquées 38 (1993), no. 6, 513–530. [21] G. L. O’Brien, Sequences of capacities, with connections to large-deviation theory, J. theoretical probab. (1996). [22] G. L. O’Brien and W. Vervaat, Capacities, large deviations and loglog laws, Stable processes and related topics (S. Cambanis, G. Samorodnitsky, and M.S. Taqqu, eds.), Progress in probability, vol. 25, Birkhaüser, 1991, pp. 43–83. [23] A. Prékopa, Logarithmic concave measures with application to stochastic programming, Acta Sci. Math. 32 (1971), 301–315. [24]. , On logarithmic concave measures and functions, Acta Sci. Math. 34 (1972), 335–343.. INRIA.

(92) On the continuity of the Cramer transform. 19. [25] J. P. Quadrat, Théorèmes asymptotiques en programmation dynamique, Note CRAS 311 (1990), 745–748. [26] R. T. Rockafellar, Convex analysis, Princeton University Press Princeton, N.J., 1970. [27] S. R. S. Varadhan, Large deviations and applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 46, SIAM, Philadelphia, Penn., 1984.. RR n ˚ 2841.

(93) Unité de recherche INRIA Lorraine, Technopôle de Nancy-Brabois, Campus scientifique, ` NANCY 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES Unité de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unité de recherche INRIA Rhône-Alpes, 46 avenue Félix Viallet, 38031 GRENOBLE Cedex 1 Unité de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex Unité de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex. ´ Editeur INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) ISSN 0249-6399.

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