[On the approximation of convex functions with cumulant generating functions]
In this Note we show that any convex function f on the real line or an interval thereof can be approximated in the norm by the cumulant generating function of a non-negative measure with an error bounded by an absolute constant which does not depend on f. We give upper and lower bounds on the best of such constants, which equal and , respectively. The proofs for these bounds are constructive. We also show that the approximation error in the multi-dimensional case is not bounded.
Nous montrons que chaque fonction convexe f définie sur la droite réelle ou un intervalle réel peut être approximée dans la norme par la fonction génératrice des cumulants d'une mésure non-négative avec une erreur bornée par une constante absolue, qui ne dépend pas de f. Nous fournissons des bornes supérieures et inférieures sur la meilleure de telles constantes, notamment et . La déduction de ces bornes est constructive. Nous montrons également que dans le cas multi-dimensionnel l'erreur de l'approximation n'est pas bornée.
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Roland Hildebrand 1
@article{CRMATH_2006__343_8_545_0, author = {Roland Hildebrand}, title = {Sur l'approximation des fonctions convexes par les fonctions g\'en\'eratrices des cumulants}, journal = {Comptes Rendus. Math\'ematique}, pages = {545--550}, publisher = {Elsevier}, volume = {343}, number = {8}, year = {2006}, doi = {10.1016/j.crma.2006.09.008}, language = {fr}, }
Roland Hildebrand. Sur l'approximation des fonctions convexes par les fonctions génératrices des cumulants. Comptes Rendus. Mathématique, Volume 343 (2006) no. 8, pp. 545-550. doi : 10.1016/j.crma.2006.09.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.09.008/
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⁎ This paper presents research results of the Belgian Programme on Interuniversity Poles of Attraction, Phase V, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture, and of the Action Concertée Incitative ‘Masses de données’ of CNRS, France. The scientific responsibility rests with its author.
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