Berry–Esseen type bounds for the matrix coefficients and the spectral radius of the left random walk on $GL_d({\protect \mathbb{R}})$

Cuny, Christophe; Dedecker, Jérôme; Merlevède, Florence; Peligrad, Magda

doi:10.5802/crmath.312

Geodesic

Parcourir par

Probabilités

Berry–Esseen type bounds for the matrix coefficients and the spectral radius of the left random walk on

G L_{d} (ℝ)

[Inégalités de type Berry–Esseen pour les coefficients matriciels et pour le rayon spectral de la marche aléatoire gauche sur

G L_{d} (ℝ)

]

Cuny, Christophe ¹ ; Dedecker, Jérôme ² ; Merlevède, Florence ³ ; Peligrad, Magda ⁴

¹ Univ. Brest, LMBA, UMR 6205 CNRS, 6 avenue Victor Le Gorgeu, 29238 Brest, France
² Université Paris Cité, MAP5, UMR 8145 CNRS, 45 rue des Saints-Pères, F-75006 Paris, France
³ Univ. Gustave Eiffel, Univ. Paris Est Créteil, UMR 8050 CNRS, LAMA, F-77454 Marne-la-Vallée, France
⁴ Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, Oh 45221-0025, USA

Comptes Rendus. Mathématique, Tome 360 (2022) no. G5, pp. 475-482.

Voir la notice de l'article provenant de la source Numdam

Abstract (VO)
Résumé

We give rates of convergence in the Central Limit Theorem for the matrix coefficients and the spectral radius of the left random walk on $G L_{d} (ℝ)$ , assuming the existence of an exponential or polynomial moment.

Nous donnons des vitesses de convergence dans le théorème limite central pour les coefficients matriciels et pour le rayon spectral de la marche aléatoire gauche sur $G L_{d} (ℝ)$ , en supposant l’existence d’un moment exponentiel ou polynomial.

Reçu le : 2021-10-18
Révisé le : 2021-12-08
Accepté le : 2021-12-09
Publié le : 2022-05-23

DOI : 10.5802/crmath.312

Classification : 60F05, 60B15, 60G50

Affiliations des auteurs :

Cuny, Christophe ¹ ; Dedecker, Jérôme ² ; Merlevède, Florence ³ ; Peligrad, Magda ⁴

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Cuny, Christophe and Dedecker, J\'er\^ome and Merlev\`ede, Florence and Peligrad, Magda},
     title = {Berry{\textendash}Esseen type bounds for the matrix coefficients and the spectral radius of the left random walk on $GL_d({\protect \mathbb{R}})$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {475--482},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
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     year = {2022},
     doi = {10.5802/crmath.312},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/articles/10.5802/crmath.312/}
}

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Cuny, Christophe; Dedecker, Jérôme; Merlevède, Florence; Peligrad, Magda. Berry–Esseen type bounds for the matrix coefficients and the spectral radius of the left random walk on $GL_d({\protect \mathbb{R}})$. Comptes Rendus. Mathématique, Tome 360 (2022) no. G5, pp. 475-482. doi : 10.5802/crmath.312. https://geodesic-test.mathdoc.fr/articles/10.5802/crmath.312/

Bibliographie
Cité par

[1] Aoun, Richard The central limit theorem for eigenvalues, Proc. Am. Math. Soc., Volume 149 (2021) no. 2, pp. 859-873 | DOI | MR | Zbl

[2] Benoist, Yves; Quint, Jean-François Central limit theorem for linear groups, Ann. Probab., Volume 44 (2016) no. 2, pp. 1308-1340 | MR | Zbl

[3] Benoist, Yves; Quint, Jean-François Random walks on reductive groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 62, Springer, 2016 | DOI | Zbl

[4] Bougerol, Philippe; Lacroix, Jean Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, 8, Birkhäuser, 1985 | DOI | Zbl

[5] Cuny, Christophe; Dedecker, Jérôme; Jan, Christophe Limit theorems for the left random walk on ${G L}_{d} (ℝ)$ , Ann. Inst. Henri Poincaré, Probab. Stat., Volume 53 (2017) no. 4, pp. 1839-1865 | MR | Zbl

[6] Cuny, Christophe; Dedecker, Jérôme; Merlevède, Florence Large and moderate deviations for the left random walk on ${G L}_{d} (ℝ)$ , ALEA, Lat. Am. J. Probab. Math. Stat., Volume 14 (2017) no. 1, pp. 503-527 | DOI | Zbl

[7] Cuny, Christophe; Dedecker, Jérôme; Merlevède, Florence; Peligrad, Magda Berry–Esseen type bounds for the left random walk on ${G L}_{d} (ℝ)$ under polynomial moment conditions (2021) (https://hal.archives-ouvertes.fr/hal-03329189/document)

[8] Dinh, Tien-Cuong; Kaufmann, Lucas; Wu, Hao Berry–Esseen bound and local limit theorem for the coefficients of products of random matrices (2021) (https://arxiv.org/abs/2110.09032)

[9] Dinh, Tien-Cuong; Kaufmann, Lucas; Wu, Hao Random walks on SL $_{2} (ℂ)$ : spectral gap and local limit theorems (2021) (https://arxiv.org/abs/2106.04019)

[10] Furstenberg, Hillel; Kesten, Harry Products of Random Matrices, Ann. Math. Stat., Volume 31 (1960) no. 2, pp. 457-469 | DOI | MR | Zbl

[11] Guivarc’h, Yves Produits de matrices aléatoires et applications aux propriétés géométriques des sous-groupes du groupe linéaire, Ergodic Theory Dyn. Syst., Volume 10 (1990) no. 3, pp. 483-512 | DOI | Zbl

[12] Hörmann, Siegfried Berry–Esseen bounds for econometric time series, ALEA, Lat. Am. J. Probab. Math. Stat., Volume 6 (2009), pp. 377-397 | MR | Zbl

[13] Le Page, Emile Théorèmes limites pour les produits de matrices aléatoires, Probability measures on groups (Oberwolfach, 1981) (Lecture Notes in Mathematics), Volume 928, Springer, 1982, pp. 258-303 | DOI | Zbl

[14] Xiao, Hui; Grama, Ion; Liu, Quansheng Berry–Esseen bounds and moderate deviations for the random walk on ${G L}_{d} (ℝ)$ , Stochastic Processes Appl., Volume 142 (2021), pp. 293-318 | Zbl | DOI

[15] Xiao, Hui; Grama, Ion; Liu, Quansheng Limit theorems for the coefficients of random walks on the general linear group (2021) (https://hal.archives-ouvertes.fr/hal-03438876/document)

Xiao, Hui; Grama, Ion; Liu, Quansheng Edgeworth expansion and large deviations for the coefficients of products of positive random matrices, Journal of Theoretical Probability, Volume 38 (2025) no. 2, p. 54 (Id/No 38) | DOI:10.1007/s10959-025-01406-z | Zbl:8005195
Christophe, Cuny Jérôme Dedecker Florence Merlevède Limit theorems for iid products of positive matrices, ALEA. Latin American Journal of Probability and Mathematical Statistics, Volume 21 (2024) no. 2, pp. 1495-1525 | DOI:10.30757/alea.v21-56 | Zbl:1551.60042
Cuny, Christophe; Dedecker, Jérôme; Merlevède, Florence Strong approximations for a class of dependent random variables with semi-exponential tails, Journal of Theoretical Probability, Volume 37 (2024) no. 3, pp. 2234-2252 | DOI:10.1007/s10959-023-01306-0 | Zbl:7900847
Dinh, Tien-Cuong; Kaufmann, Lucas; Wu, Hao Berry-Esseen bound and local limit theorem for the coefficients of products of random matrices, Journal of the Institute of Mathematics of Jussieu, Volume 23 (2024) no. 2, pp. 705-735 | DOI:10.1017/s1474748022000561 | Zbl:1543.60008
Dinh, Tien-Cuong; Kaufmann, Lucas; Wu, Hao Random walks on ${SL}_{2} (C)$ : spectral gap and limit theorems, Probability Theory and Related Fields, Volume 186 (2023) no. 3-4, pp. 877-955 | DOI:10.1007/s00440-023-01191-y | Zbl:1517.60030
Xiao, Hui; Grama, Ion; Liu, Quansheng Moderate deviations and local limit theorems for the coefficients of random walks on the general linear group, Stochastic Processes and their Applications, Volume 158 (2023), pp. 103-133 | DOI:10.1016/j.spa.2022.12.013 | Zbl:1510.60020
Cuny, C.; Dedecker, J.; Merlevède, F.; Peligrad, M. Berry-Esseen type bounds for the left random walk on ${GL}_{d} (R)$ under polynomial moment conditions, The Annals of Probability, Volume 51 (2023) no. 2, pp. 495-523 | DOI:10.1214/22-aop1602 | Zbl:1519.60031
Dinh, Tien-Cuong; Kaufmann, Lucas; Wu, Hao Berry-Esseen bounds with targets and local limit theorems for products of random matrices, The Journal of Geometric Analysis, Volume 33 (2023) no. 3, p. 42 (Id/No 76) | DOI:10.1007/s12220-022-01127-3 | Zbl:1506.60012

Cité par 8 documents. Sources : zbMATH