[Historique et avancées récentes sur le théorème de la limite locale]
En théorie des probabilités, les théorèmes de la limite locale donnent un développement asymptotique des puissances itérées d’une loi de probabilité supportée sur avec des estimations uniformes sur les termes de reste. Dans cette revue, nous présentons des résultats récents pour les convolutions itérées de suites complexes intégrables en une dimension d’espace. Dans le cas parabolique, nous donnons un développement complet, à tous les ordres, et nous obtenons une estimation ponctuelle précise des termes de reste sous forme de Gaussiennes généralisées. Nous présentons également une extension de notre résultat principal dans le cas semi-discret (problèmes continus en temps de convolution), et nous discutons plusieurs perspectives naturelles à ce travail.
In probability theory, local limit theorems provide an asymptotic expansion of the convolution powers of a probability distribution supported on with uniform bounds on the remainders. In this review, we present some recent results for the iterated convolution of complex valued integrable sequences in one space dimension. In the so-called parabolic case, we give a complete expansion, at any accuracy order, for these convolution powers and we provide sharp, pointwise, generalized Gaussian bounds for the remainders. We also present an extension of our main result to the semi-discrete setting (time-continuous convolution problems), and discuss several natural perspectives.
Keywords: Convolution, asymptotic expansion, stability, local limit theorem
Mot clés : Convolution, développement asymptotique, stabilité, théorème de la limite locale
@incollection{JEDP_2024____A5_0,
author = {Jean-Fran\c{c}ois Coulombel and Gr\'egory Faye},
title = {Local limit theorems for complex valued sequences: {Old} & {New}},
booktitle = {},
series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
note = {talk:5},
pages = {1--15},
publisher = {R\'eseau th\'ematique AEDP du CNRS},
year = {2024},
doi = {10.5802/jedp.686},
language = {en},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.686/}
}
TY - JOUR AU - Jean-François Coulombel AU - Grégory Faye TI - Local limit theorems for complex valued sequences: Old & New JO - Journées équations aux dérivées partielles N1 - talk:5 PY - 2024 SP - 1 EP - 15 PB - Réseau thématique AEDP du CNRS UR - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.686/ DO - 10.5802/jedp.686 LA - en ID - JEDP_2024____A5_0 ER -
%0 Journal Article %A Jean-François Coulombel %A Grégory Faye %T Local limit theorems for complex valued sequences: Old & New %J Journées équations aux dérivées partielles %Z talk:5 %D 2024 %P 1-15 %I Réseau thématique AEDP du CNRS %U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.686/ %R 10.5802/jedp.686 %G en %F JEDP_2024____A5_0
Jean-François Coulombel; Grégory Faye. Local limit theorems for complex valued sequences: Old & New. Journées équations aux dérivées partielles (2024), Exposé no. 5, 15 p. doi : 10.5802/jedp.686. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.686/
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