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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Abstract

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.

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773728

Digital Object Identifier
doi:10.1214/16-AIHP773

Mathematical Reviews number (MathSciNet)
MR3729637

Zentralblatt MATH identifier
06847064

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

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

Citation

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. https://projecteuclid.org/euclid.aihp/1511773728


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References

  • [1] G. Alsmeyer. Convergence rates in the law of large numbers for martingales. Stochastic Process. Appl. 36 (2) (1990) 181–194.
  • [2] Y. Benoist and J.-F. Quint. Central limit theorem for linear groups. Ann. Probab. 44 (2) (2016) 1308–1340.
  • [3] Y. Benoist and J.-F. Quint Random walks on reductive groups manuscript. Available at http://www.math.u-psud.fr/~benoist/prepubli/prepublication.html.
  • [4] I. Berkes, W. Liu and W. B. Wu. Komlós–Major–Tusnády approximation under dependence. Ann. Probab. 42 (2) (2014) 794–817.
  • [5] P. Bougerol and J. Lacroix. Products of Random Matrices with Applications to Schrödinger Operators. Progress in Probability and Statistics 8. Birkhäuser, Boston, MA, 1985.
  • [6] C. Cuny. ASIP for martingales in 2-smooth Banach spaces. Applications to stationary processes. Bernoulli 21 (1) (2015) 374–400.
  • [7] C. Cuny. Invariance principles under the Maxwell–Woodroofe condition in Banach spaces. Ann. Probab., 2016. To appear. Available at arXiv:1403.0772.
  • [8] C. Cuny and F. Merlevède. Strong invariance principles with rate for “reverse” martingale differences and applications. J. Theoret. Probab. 28 (1) (2015) 137–183.
  • [9] J. Dedecker, P. Doukhan and F. Merlevède. Rates of convergence in the strong invariance principle under projective criteria. Electron. J. Probab. 17 (2012) 16. DOI:10.1214/EJP.v17-1849.
  • [10] J. Dedecker, F. Merlevède and E. Rio. Rates of convergence for minimal distances in the central limit theorem under projective criteria. Electron. J. Probab. 14 (35) (2009) 978–1011.
  • [11] J. Dedecker and E. Rio. On the functional central limit theorem for stationary processes. Ann. Inst. Henri Poincaré Probab. Stat. 36 (2000) 1–34.
  • [12] H. Furstenberg and H. Kesten. Products of random matrices. Ann. Math. Stat. 31 (1960) 457–469.
  • [13] Y. Guivarc’h and A. Raugi. Frontière de furstenberg, propriétés de contraction et théorèmes de convergence. Z. Wahrsch. Verw. Gebiete 69 (2) (1985) 187–242.
  • [14] S. Hao and Q. Liu. Convergence rates in the law of large numbers for arrays of martingale differences. J. Math. Anal. Appl. 417 (2) (2014) 733–773.
  • [15] C. C. Heyde. On the central limit theorem and iterated logarithm law for stationary processes. Bull. Aust. Math. Soc. 12 (1975) 1–8.
  • [16] C. Jan. Vitesse de convergence dans le TCL pour des chaînes de Markov et certains processus associés à des systèmes dynamiques. C. R. Acad. Sci. Paris Sér. I Math. 331 (5) (2000) 395–398.
  • [17] C. Jan. Vitesse de convergence dans le TCL pour des processus associés à des systèmes dynamiques ou des produits de matrices aléatoires. Thèse de l’université de Rennes, 1. Thesis number 01REN10073, 2001.
  • [18] J. Komlós, P. Major and G. Tusnády. An approximation of partial sums of independent RV’s, and the sample DF. I. Z. Warsch. Verw. Gebiete 32 (1975) 111–131.
  • [19] U. Krengel. Ergodic Theorems. De Gruyter Studies in Mathematics 6. Walter de Gruyter, Berlin, 1985.
  • [20] E. Le Page. Théorèmes limites pour les produits de matrices aléatoires. In Probability Measures on Groups 258–303. Oberwolfach, 1981. Lecture Notes in Math. 928. Springer, Berlin, 1982.
  • [21] M. Maxwell and M. Woodroofe. Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 (2) (2000) 713–724.
  • [22] F. Merlevède and M. Peligrad. Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples. Ann. Probab. 41 (2) (2013) 914–960.
  • [23] M. Peligrad and S. Utev. A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33 (2) (2005) 798–815.
  • [24] M. Peligrad, S. Utev and W. B. Wu. A maximal lp-inequality for stationary sequences and its applications. Proc. Amer. Math. Soc. 135 (2) (2007) 541–550.
  • [25] Q. M. Shao. Almost sure invariance principles for mixing sequences of random variables. Stochastic Process. Appl. 48 (1993) 319–334.
  • [26] B. von Bahr and C.-G. Esseen. Inequalities for the $r$th absolute moment of a sum of random variables, $1\le r\le2$. Ann. Math. Stat. 36 (1965) 299–303.
  • [27] W. B. Wu and Z. Zhao. Moderate deviations for stationary processes. Statist. Sinica 18 (2) (2008) 769–782.