Comptes Rendus
Research article - Numerical analysis
The local limit theorem for complex valued sequences: the parabolic case
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1801-1818.

We give a complete expansion, at any accuracy order, for the iterated convolution of a complex valued integrable sequence in one space dimension. The remainders are estimated sharply with generalized Gaussian bounds. The result applies in probability theory for random walks as well as in numerical analysis for studying the large time behavior of numerical schemes.

Nous donnons un développement asymptotique à tout ordre pour la convolution itérée d’une suite complexe intégrable en une dimension d’espace. Les restes sont estimés de manière optimale avec une borne Gaussienne généralisée. Le résultat s’applique tant en théorie des probabilités pour les marches aléatoires qu’en analyse numérique pour le comportement en temps grand de schémas numériques aux différences finies.

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DOI: 10.5802/crmath.685
Classification: 42A85, 35K25, 60F99, 65M12
Keywords: Convolution, asymptotic expansion, stability, local limit theorem
Mots-clés : Convolution, développement asymptotique, stabilité, théorème de la limite locale

Jean-François Coulombel 1; Grégory Faye 1

1 Institut de Mathématiques de Toulouse – UMR 5219, Université de Toulouse, CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9 , France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {The local limit theorem for complex valued sequences: the parabolic case},
     journal = {Comptes Rendus. Math\'ematique},
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Jean-François Coulombel; Grégory Faye. The local limit theorem for complex valued sequences: the parabolic case. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1801-1818. doi : 10.5802/crmath.685. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.685/

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