Sur l'approximation des fonctions convexes par les fonctions génératrices des cumulants

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  • C. R. Acad. Sci. Paris, Ser. I 343 (2006) 545550


    Sur lapproximation des fonctions convexes par les fonctionsgnratrices des cumulants

    Roland HildebrandLMC, universit Joseph-Fourier, tour IRMA, 51, rue des mathmatiques, 38400 St. Martin dHres, France

    Reu le 7 octobre 2004 ; accept aprs rvision le 5 septembre 2006

    Disponible sur Internet le 12 octobre 2006

    Prsent par Paul Deheuvels


    Nous montrons que chaque fonction convexe f dfinie sur la droite relle ou un intervalle rel peut tre approxime dans lanorme C0 par la fonction gnratrice des cumulants dune msure non-ngative avec une erreur borne par une constante absolue,qui ne dpend pas de f . Nous fournissons des bornes suprieures et infrieures sur la meilleure de telles constantes, notammentln 2 et ln 22 . La dduction de ces bornes est constructive. Nous montrons galement que dans le cas multi-dimensionnel lerreur delapproximation nest pas borne. Pour citer cet article : R. Hildebrand, C. R. Acad. Sci. Paris, Ser. I 343 (2006). 2006 Acadmie des sciences. Publi par Elsevier Masson SAS. Tous droits rservs.Abstract

    On the approximation of convex functions with cumulant generating functions. In this Note we show that any convexfunction f on the real line or an interval thereof can be approximated in the C0 norm by the cumulant generating function of anon-negative measure with an error bounded by an absolute constant which does not depend on f . We give upper and lower boundson the best of such constants, which equal ln 2 and ln 22 , respectively. The proofs for these bounds are constructive. We also showthat the approximation error in the multi-dimensional case is not bounded. To cite this article: R. Hildebrand, C. R. Acad. Sci.Paris, Ser. I 343 (2006). 2006 Acadmie des sciences. Publi par Elsevier Masson SAS. Tous droits rservs.

    Abridged English version

    Let f (t) be a real-valued measurable function defined on the real line. The two-sided Laplace transform of f isdefined as F(s) = + est f (t)dt . It is well known that the region of convergence of this integral is a vertical strip inthe complex plane, which may be unbounded to the left or to the right or to both sides, and in this region the functionF(s) is analytic [6].

    This paper presents research results of the Belgian Programme on Interuniversity Poles of Attraction, Phase V, initiated by the Belgian State,Prime Ministers Office for Science, Technology and Culture, and of the Action Concerte Incitative Masses de donnes of CNRS, France. Thescientific responsibility rests with its author.

    Adresse e-mail : (R. Hildebrand).1631-073X/$ see front matter 2006 Acadmie des sciences. Publi par Elsevier Masson SAS. Tous droits rservs.doi:10.1016/j.crma.2006.09.008

  • 546 R. Hildebrand / C. R. Acad. Sci. Paris, Ser. I 343 (2006) 545550Let (t) be a probability measure and M(s) its two-sided Laplace transform. Then the function M(s) = e

    st(t)dt generates the moments of this measure. Its logarithm (s) = lnM(s) generates the cumulants ofthe probability measure and is thus called cumulant generating function (CGF) [5]. Since is a probability measure,the cumulant generating function satisfies the relation (0) = 0. To avoid this restriction, we further consider CGFsof unnormalized non-negative measures, which not necessarily sum up to 1. In the sequel, if we speak of a CGFwe assume the CGF of an unnormalized non-negative measure. CGFs have many applications in statistics, e.g. theydescribe the asymptotics of large deviations [4] and play a central role in the apparatus of the exponential family ofdistributions [3].

    A CGF (s) of a non-negative measure (t) is an analytic function whose derivatives satisfy certain polynomial in-equalities. To obtain these inequalities, recall that a sequence of real numbers {m0,m1, . . . ,mn, . . .} can be representedas the sequence of moments of a non-negative measure if and only if all Hankel matrices of the form

    H(m0,m1, . . . ,m2n) =

    m0 m1 mnm1 m2 mn+1...

    mn mn+1 m2n

    are positive semidefinite [2]. This yields the non-negativity of the principal minors of these matrices. These minorsare polynomials in the mk . But mk is the k-th order derivative at zero of the exponent e(s) of the CGF of the measureand hence can be expressed through the derivatives of (s) at zero. In this way one obtains a sequence of inequalitieson the derivatives of (s), which can easily be verified to be also polynomial. By multiplication of the measure withes0t one obtains the same inequalities for the derivatives at an arbitrary point s0.

    The first two non-trivial inequalities are m0m2 m21 0, m0m2m4 +2m1m2m3 m0m23 m4m21 m32 0, whichlead to 0, ((IV) + 2()2) ()2 0. While the first inequality involves only moments up to order 2 ofthe corresponding measure, the following inequalities involve higher-order moments. Note that the first inequalityexpresses the condition of convexity of the CGF.

    Obviously not every convex function is a CGF of some non-negative measure, already because CGFs are analyticfunctions. One can then ask how well an arbitrary convex function can be approximated by a CGF. This questionconstitutes the subject of the present note.

    The surprising answer is that a convex function can be approximated by a CGF with an error in the C0 norm thatis bounded above by an absolute constant, no matter whether the convex function is defined on a finite interval or aninfinite one or whether it is bounded. We prove an upper and a lower bound to the best such constant. These boundsequal ln 2 and ln 22 , respectively. Namely, we have the following results:

    Theorem 0.1. Let f (x) be a continuous convex function, defined on the real line or an interval of the real line. Thenthere exists a non-negative measure whose CGF (x) is defined on the same interval and f C0 ln 2.

    Theorem 0.2. Let > 0. Then there exists a continuous convex function f (x) on the interval [1,1] such that for anyCGF (x) of a non-negative measure we have f C0([1,1]) ln 22 .

    We also consider the approximation of convex functions in real spaces of higher dimensions by multivariate CGFs.The CGF of a non-negative measure (t) on Rn is defined to be the function (s) = ln

    Rnes,t(t)dt , where ,

    is the usual scalar product in Rn. This problem is the straightforward generalization of the preceding approximationproblem from the case of one to the case of several independent variables. We show that the above results cannotbe extended to the multivariable case, and there exist convex functions on Rn, n 2, for which the approximationby CGFs is arbitrarily bad. Namely, for n 2 there exist convex functions on convex domains of Rn which areapproximated by multivariate CGFs with an arbitrarily large error in the C0 norm. We have the following result:

    Theorem 0.3. Let n 2 be an integer and R > 0 a real number. Then there exists a continuous convex function f (x)on the closed ball B1 Rn with unit radius such that for any n-variate CGF (x) of a non-negative measure on B1we have f C0(B1) R.Theorems 0.10.3 will be proven in the next three sections, respectively.

  • R. Hildebrand / C. R. Acad. Sci. Paris, Ser. I 343 (2006) 545550 5471. Borne suprieure

    Thorme 1.1. Soit f (x) une fonction convexe continue, dfinie sur la droite relle ou un intervalle rel. Alors ilexiste une msure non-ngative dont la fonction gnratrice des cumulants (x) est dfinie sur le mme intervalle etf C0 ln 2.

    Preuve. Soit x0 un point dans le domaine D de dfinition de f et soit p0 un sous-gradient de f x0. Dfinissons lafonction tx0,p0(x) = f (x) f (x0) p0(x x0). Elle est non-ngative, convexe (par rapport x) et possde un zro x = x0. De plus, si x0 x1 x2 sont des points dans D et p0 p1 p2 sont des sous-gradients de f dans cespoints, alors on a les ingalits suivantes :

    tx0,p0(x2) tx0,p0(x1) + tx1,p1(x2), tx2,p2(x0) tx2,p2(x1) + tx1,p1(x0). (1)Nous allons construire dune manire itrative une squence de fonctions linaires . . . , l2(x), l1(x), l0(x),

    l1(x), l2(x), . . . sur D, o lindex k parcourt un ensemble K Z des entiers consecutifs. Lensemble K peut tre infinisur chaque ct et contient le zro. Ces fonctions obeissent lingalit lk(x) f (x) pour tout x D. Dfinissons lafonction l0(x) en fixant un point x0 D et un sous-gradient p0 de f x0 et en mettant l0(x) = f (x0) + p0(x x0).Les fonctions l(k+1)(x) seront construites partir des fonctions lk(x). Dailleurs, une de ces fonctions ou toutes lesdeux peuvent faillir dexister. Dans ce cas-l la squence se termine dans la direction correspondante. La construc-tion pour la direction positive et la direction ngative est similaire, et nous ne dcrivons que celle pour les indicesnon-ngatifs.

    Pour k = 0 lingalit l0(x) f (x) est satisfaite par la convexit de f . Soit k 0 et = ln 2. Supposons quelk(x) f (x) est dj construite et montrons la construction de la fonction linaire lk+1(x). Soit pk la drive de lk(x).Sil ny a aucun point x D tel que lk(x) = f (x), alors lk+1 nexiste pas et k est lindex maximal de lensemble K .Sil existe un tel point x D, on en choisit un, disons xk . Alors il suit de lingalit lk(x) f (x) que pk est unsous-gradient de f xk . Si on a txk,pk (x) pour tout x > xk , x D, alors lk+1 nexiste pas non plus et k estlindex maximal de K . Sil existe x D, x > xk tel que txk,pk (x) > , alors il existe un point unique x > xk tel quetxk,pk (x

    ) = . Posons pk+1 = sup{p | f (x) + p(x x) f (x) x D}. Lensemble de tels points p nestpas vide et son supremum est suprieur pk parce quil contient tous les sous-gradients de f x. Le supremum estaussi fini, parce quil existe x D, x > x, et est alors atteint. Dfinissons lk+1(x) = f (x) + pk+1(x x). Parla construction nous avons lk+1(x) f (x