[Le théorème de la limite locale pour les suites complexes : le cas parabolique]
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.
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.685
Keywords: Convolution, asymptotic expansion, stability, local limit theorem
Mots-clés : Convolution, développement asymptotique, stabilité, théorème de la limite locale
Coulombel, Jean-François 1 ; Faye, Grégory 1
CC-BY 4.0
@article{CRMATH_2024__362_G12_1801_0,
author = {Coulombel, Jean-Fran\c{c}ois and Faye, Gr\'egory},
title = {The local limit theorem for complex valued sequences: the parabolic case},
journal = {Comptes Rendus. Math\'ematique},
pages = {1801--1818},
year = {2024},
publisher = {Acad\'emie des sciences, Paris},
volume = {362},
number = {G12},
doi = {10.5802/crmath.685},
zbl = {07949988},
language = {en},
url = {https://www.numdam.org/articles/10.5802/crmath.685/}
}
TY - JOUR AU - Coulombel, Jean-François AU - Faye, Grégory TI - The local limit theorem for complex valued sequences: the parabolic case JO - Comptes Rendus. Mathématique PY - 2024 SP - 1801 EP - 1818 VL - 362 IS - G12 PB - Académie des sciences, Paris UR - https://www.numdam.org/articles/10.5802/crmath.685/ DO - 10.5802/crmath.685 LA - en ID - CRMATH_2024__362_G12_1801_0 ER -
%0 Journal Article %A Coulombel, Jean-François %A Faye, Grégory %T The local limit theorem for complex valued sequences: the parabolic case %J Comptes Rendus. Mathématique %D 2024 %P 1801-1818 %V 362 %N G12 %I Académie des sciences, Paris %U https://www.numdam.org/articles/10.5802/crmath.685/ %R 10.5802/crmath.685 %G en %F CRMATH_2024__362_G12_1801_0
Coulombel, Jean-François; Faye, Grégory. The local limit theorem for complex valued sequences: the parabolic case. Comptes Rendus. Mathématique, Tome 362 (2024) no. G12, pp. 1801-1818. doi: 10.5802/crmath.685
[1] A generalized polar-coordinate integration formula with applications to the study of convolution powers of complex-valued functions on , J. Fourier Anal. Appl., Volume 28 (2022) no. 2, 19, 74 pages | DOI | MR | Zbl
[2] Besov spaces and applications to difference methods for initial value problems, Lecture Notes in Mathematics, 434, Springer, 1975, ii+154 pages | DOI | MR | Zbl
[3] Analyse quantitative des schémas numériques pour les équations aux dérivées partielles, EDP Sciences, 2024 | DOI | Zbl
[4] Generalized Gaussian bounds for discrete convolution powers, Rev. Mat. Iberoam., Volume 38 (2022) no. 5, pp. 1553-1604 | DOI | MR | Zbl
[5] Advanced combinatorics, D. Reidel Publishing Co., 1974, xi+343 pages (The art of finite and infinite expansions) | MR | Zbl | DOI
[6] The Green’s function of the Lax–Wendroff and Beam–Warming schemes, Ann. Math. Blaise Pascal, Volume 29 (2022) no. 2, pp. 247-294 | DOI | MR | Zbl | Numdam
[7] Local limit theorem for complex valued sequences (2022) (to appear in Asymptotic Anal. https://arxiv.org/abs/2201.01514)
[8] Finite volume transport schemes, Numer. Math., Volume 108 (2008) no. 4, pp. 529-556 | DOI | MR | Zbl
[9] Uniform asymptotic stability of Strang’s explicit compact schemes for linear advection, SIAM J. Numer. Anal., Volume 47 (2009) no. 5, pp. 3956-3976 | DOI | MR | Zbl
[10] Convolution powers of complex functions on , Math. Nachr., Volume 287 (2014) no. 10, pp. 1106-1130 | DOI | MR | Zbl
[11] Norms of powers of absolutely convergent Fourier series, Mich. Math. J., Volume 13 (1966), pp. 393-416 | DOI | MR | Zbl
[12] Survey of the stability of linear finite difference equations, Commun. Pure Appl. Math., Volume 9 (1956), pp. 267-293 | DOI | MR | Zbl
[13] Sums of independent random variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, 82, Springer, 1975, x+346 pages (Translated from the Russian by A. A. Brown) | MR | Zbl
[14] Local limit theorems for complex functions on , J. Math. Anal. Appl., Volume 519 (2023) no. 2, 126832, 64 pages | DOI | MR | Zbl
[15] On the convolution powers of complex functions on , J. Fourier Anal. Appl., Volume 21 (2015) no. 4, pp. 754-798 | DOI | MR | Zbl
[16] Convolution powers of complex functions on , Rev. Mat. Iberoam., Volume 33 (2017) no. 3, pp. 1045-1121 | DOI | MR | Zbl
[17] On-diagonal asymptotics for heat kernels of a class of inhomogeneous partial differential operators, J. Differ. Equations, Volume 363 (2023), pp. 67-125 | DOI | MR | Zbl
[18] Real and complex analysis, McGraw-Hill, 1987, xiv+416 pages | Zbl | MR
[19] Trigonometric polynomials and difference methods of maximum accuracy, J. Math. Phys., Volume 41 (1962), pp. 147-154 | DOI | Zbl
[20] Orthogonal polynomials, Colloquium Publications, XXIII, American Mathematical Society, 1975, xiii+432 pages | MR | Zbl
[21] Stability of difference schemes in the maximum-norm, J. Differ. Equations, Volume 1 (1965), pp. 273-292 | DOI | MR | Zbl
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