[Le théorème de la limite locale pour les suites complexes : le cas parabolique]
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.
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.
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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
CC-BY 4.0
@article{CRMATH_2024__362_G12_1801_0,
author = {Jean-Fran\c{c}ois Coulombel and Gr\'egory Faye},
title = {The local limit theorem for complex valued sequences: the parabolic case},
journal = {Comptes Rendus. Math\'ematique},
pages = {1801--1818},
publisher = {Acad\'emie des sciences, Paris},
volume = {362},
year = {2024},
doi = {10.5802/crmath.685},
language = {en},
}
TY - JOUR AU - Jean-François Coulombel AU - Grégory Faye TI - The local limit theorem for complex valued sequences: the parabolic case JO - Comptes Rendus. Mathématique PY - 2024 SP - 1801 EP - 1818 VL - 362 PB - Académie des sciences, Paris DO - 10.5802/crmath.685 LA - en ID - CRMATH_2024__362_G12_1801_0 ER -
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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