Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Limit theorems for the left random walk on $\operatorname{GL}_{d}(\mathbb{R})$

Christophe Cuny, Jérôme Dedecker, and Christophe Jan

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Motivated by a recent work of Benoist and Quint and extending results from the PhD thesis of the third author, we obtain limit theorems for products of independent and identically distributed elements of $\operatorname{GL}_{d}(\mathbb{R})$, such as the Marcinkiewicz–Zygmund strong law of large numbers, the CLT (with rates in Wasserstein’s distances) and almost sure invariance principles with rates.


Motivés par un travail récent de Benoist et Quint, nous étendons certains résultats issus de la thèse de doctorat du troisième auteur puis établissons des théorèmes limite pour les produits de matrices indépendantes et identiquement distribuées de $\operatorname{GL}_{d}(\mathbb{R})$. Nous nous intéressons notamment aux lois fortes de Marcinkiewicz–Zygmund, au TLC (avec vitesses en distance de Wasserstein) et au principe d’invariance presque-sûr avec vitesse.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 1839-1865.

Received: 3 March 2016
Accepted: 9 June 2016
First available in Project Euclid: 27 November 2017

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Zentralblatt MATH identifier

Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles

Central Limit theorem Random walks on $\operatorname{GL}_{d}(\mathbb{R})$ Strong invariance principles


Cuny, Christophe; Dedecker, Jérôme; Jan, Christophe. Limit theorems for the left random walk on $\operatorname{GL}_{d}(\mathbb{R})$. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 1839--1865. doi:10.1214/16-AIHP773.

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