Résumé : The class of completely monotone functions and Bernstein functions is important in the context of infinitely divisible probability measures of $(0,\infty)$. We shall provide for them several new characterizations via two approaches: the first one is driven algebraically via elementary preserving mappings and the second one is developed in terms of the behavior of their restriction on $N$. As a consequence, we give a complete answer to the following question: "Can we affirm that a function $f$ is completely monotone (resp. a Bernstein function) if we know that the sequence $(f (k))_k$ is completely monotone (resp. alternating)?". This approach constitutes a kind of converse to Hausdorff’s moment characterization theorem in the context of completely monotone sequences. With closely related tools, we solve an open problem raised by Harkness and Shantaram (1969) who obtained, under sufficient conditions, a limit theorem in law for sequences of nonnegative random variables build with the iterated stationary excess operator. We show that the conditions of Harkness and Shantaram are actually necessary; continuous time convergence is equivalent to discrete time convergence; and the only possible limits in distribution are mixture of exponential with log-normal distributions.
[NB: For this talk, we have the pleasure to join the "Applied Mathematics Webinar", Jeddah - Riadh - Dammam - Tunis]
Dernière modification : Tuesday 03 November 2020 | Contact : Cyril.Banderier at lipn.univ-paris13.fr |