Rates of convergence in invariance principles for random walks on linear groups via martingale methods
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- by C. Cuny, J. Dedecker and F. Merlevède PDF
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Abstract:
In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general ${\mathbb R}^d$-valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates either in the central limit theorem for the Wasserstein distance, or in the strong approximation. A part of this paper is devoted to circumvent this issue. We then exhibit near optimal rates both in the central limit theorem in terms of the Wasserstein distance and in the almost sure invariance principle for ${\mathbb R}^d$-valued martingales with stationary increments having moments of order $p \in (2, 3]$ (the case of sequences of reversed martingale differences is also considered). Note also that, as an application of our results for general ${\mathbb R}^d$-valued cocycles, a special attention is paid to the Iwasawa cocycle and the Cartan projection for reductive Lie groups (like for instance ${\mathrm {GL}}_d(\mathbb {R})$).References
- Yves Benoist and Jean-François Quint, Central limit theorem for linear groups, Ann. Probab. 44 (2016), no. 2, 1308–1340. MR 3474473, DOI 10.1214/15-AOP1002
- I. Benua and Zh. -F. Kén, Central limit theorem on hyperbolic groups, Izv. Ross. Akad. Nauk Ser. Mat. 80 (2016), no. 1, 5–26 (Russian, with Russian summary); English transl., Izv. Math. 80 (2016), no. 1, 3–23. MR 3462675, DOI 10.4213/im8306
- Yves Benoist and Jean-François Quint, Random walks on reductive groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 62, Springer, Cham, 2016. MR 3560700, DOI 10.1007/978-3-319-47721-3
- István Berkes, Weidong Liu, and Wei Biao Wu, Komlós-Major-Tusnády approximation under dependence, Ann. Probab. 42 (2014), no. 2, 794–817. MR 3178474, DOI 10.1214/13-AOP850
- Michael Björklund, Central limit theorems for Gromov hyperbolic groups, J. Theoret. Probab. 23 (2010), no. 3, 871–887. MR 2679960, DOI 10.1007/s10959-009-0230-x
- Philippe Bougerol and Jean Lacroix, Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 886674, DOI 10.1007/978-1-4684-9172-2
- Christophe Cuny, Jérôme Dedecker, and Christophe Jan, Limit theorems for the left random walk on $\textrm {GL}_d(\Bbb R)$, Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017), no. 4, 1839–1865 (English, with English and French summaries). MR 3729637, DOI 10.1214/16-AIHP773
- Christophe Cuny, Jérôme Dedecker, and Florence Merlevède, Large and moderate deviations for the left random walk on $GL_d(\Bbb R)$, ALEA Lat. Am. J. Probab. Math. Stat. 14 (2017), no. 1, 503–527. MR 3659265, DOI 10.30757/alea.v14-26
- Christophe Cuny, Jérôme Dedecker, and Florence Merlevède, On the Komlós, Major and Tusnády strong approximation for some classes of random iterates, Stochastic Process. Appl. 128 (2018), no. 4, 1347–1385. MR 3769665, DOI 10.1016/j.spa.2017.07.011
- Christophe Cuny and Florence Merlevède, Strong invariance principles with rate for “reverse” martingale differences and applications, J. Theoret. Probab. 28 (2015), no. 1, 137–183. MR 3320963, DOI 10.1007/s10959-013-0506-z
- Jérôme Dedecker, Florence Merlevède, and Emmanuel Rio, Rates of convergence for minimal distances in the central limit theorem under projective criteria, Electron. J. Probab. 14 (2009), no. 35, 978–1011. MR 2506123, DOI 10.1214/EJP.v14-648
- Jérôme Dedecker, Florence Merlevède, and Emmanuel Rio, Strong approximation results for the empirical process of stationary sequences, Ann. Probab. 41 (2013), no. 5, 3658–3696. MR 3127895, DOI 10.1214/12-AOP798
- Jérôme Dedecker, Clémentine Prieur, and Paul Raynaud De Fitte, Parametrized Kantorovich-Rubinštein theorem and application to the coupling of random variables, Dependence in probability and statistics, Lect. Notes Stat., vol. 187, Springer, New York, 2006, pp. 105–121. MR 2283252, DOI 10.1007/0-387-36062-X_{5}
- Yves Derriennic and Michael Lin, Convergence of iterates of averages of certain operator representations and of convolution powers, J. Funct. Anal. 85 (1989), no. 1, 86–102. MR 1005857, DOI 10.1016/0022-1236(89)90047-5
- Ernst Eberlein, On strong invariance principles under dependence assumptions, Ann. Probab. 14 (1986), no. 1, 260–270. MR 815969
- Xiequan Fan, Ion Grama, and Quansheng Liu, Deviation inequalities for martingales with applications, J. Math. Anal. Appl. 448 (2017), no. 1, 538–566. MR 3579898, DOI 10.1016/j.jmaa.2016.11.023
- D. H. Fuk, Certain probabilistic inequalities for martingales, Sibirsk. Mat. Ž. 14 (1973), 185–193, 239 (Russian). MR 0326835
- H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist. 31 (1960), 457–469. MR 121828, DOI 10.1214/aoms/1177705909
- Sébastien Gouëzel, Almost sure invariance principle for dynamical systems by spectral methods, Ann. Probab. 38 (2010), no. 4, 1639–1671. MR 2663640, DOI 10.1214/10-AOP525
- Y. Guivarc’h and A. Raugi, Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Z. Wahrsch. Verw. Gebiete 69 (1985), no. 2, 187–242 (French). MR 779457, DOI 10.1007/BF02450281
- Camille Horbez, Central limit theorems for mapping class groups and $\textrm {Out}(F_N)$, Geom. Topol. 22 (2018), no. 1, 105–156. MR 3720342, DOI 10.2140/gt.2018.22.105
- Yukiyosi Kawada and Kiyosi Itô, On the probability distribution on a compact group. I, Proc. Phys.-Math. Soc. Japan (3) 22 (1940), 977–998. MR 3462
- 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 (2001), thesis number 01REN10073.
- S. Karmakar and W. B. Wu, Optimal Gaussian approximation for multiple times series, Statistica Sinica 30 (2020), 1399–1417. DOI: 10.5705/ss.202017.0303
- A. Korepanov, Z. Kosloff, and I. Melbourne, Martingale-coboundary decomposition for families of dynamical systems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 4, 859–885. MR 3795019, DOI 10.1016/j.anihpc.2017.08.005
- Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23, Springer-Verlag, Berlin, 1991. Isoperimetry and processes. MR 1102015, DOI 10.1007/978-3-642-20212-4
- Émile Le Page, Théorèmes limites pour les produits de matrices aléatoires, Probability measures on groups (Oberwolfach, 1981) Lecture Notes in Math., vol. 928, Springer, Berlin-New York, 1982, pp. 258–303 (French). MR 669072
- Ian Melbourne and Matthew Nicol, A vector-valued almost sure invariance principle for hyperbolic dynamical systems, Ann. Probab. 37 (2009), no. 2, 478–505. MR 2510014, DOI 10.1214/08-AOP410
- Florence Merlevède, Magda Peligrad, and Sergey Utev, Functional Gaussian approximation for dependent structures, Oxford Studies in Probability, vol. 6, Oxford University Press, Oxford, 2019. MR 3930596, DOI 10.1093/oso/9780198826941.001.0001
- Florence Merlevède and Emmanuel Rio, Strong approximation of partial sums under dependence conditions with application to dynamical systems, Stochastic Process. Appl. 122 (2012), no. 1, 386–417. MR 2860454, DOI 10.1016/j.spa.2011.08.012
- D. Monrad and W. Philipp, The problem of embedding vector-valued martingales in a Gaussian process, Teor. Veroyatnost. i Primenen. 35 (1990), no. 2, 384–387; English transl., Theory Probab. Appl. 35 (1990), no. 2, 374–377 (1991). MR 1069143, DOI 10.1137/1135050
- Gregory Morrow and Walter Philipp, An almost sure invariance principle for Hilbert space valued martingales, Trans. Amer. Math. Soc. 273 (1982), no. 1, 231–251. MR 664040, DOI 10.1090/S0002-9947-1982-0664040-3
- Françoise Pène, Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard, Ann. Appl. Probab. 15 (2005), no. 4, 2331–2392. MR 2187297, DOI 10.1214/105051605000000476
- Ludger Rüschendorf, The Wasserstein distance and approximation theorems, Z. Wahrsch. Verw. Gebiete 70 (1985), no. 1, 117–129. MR 795791, DOI 10.1007/BF00532240
- A. V. Skorohod, On a representation of random variables, Teor. Verojatnost. i Primenen. 21 (1976), no. 3, 645–648 (Russian, with English summary). MR 0428369
Additional Information
- C. Cuny
- Affiliation: UMR CNRS 6205, Laboratoire de Mathématiques de Bretagne Atlantique, Univ Brest, 6 av. Le Gorgeu, 29238 Brest, France
- MR Author ID: 670575
- Email: christophe.cuny@univ-brest.fr
- J. Dedecker
- Affiliation: Université de Paris, CNRS, MAP5, UMR 8145, 45 rue des Saints Pères, 75006 Paris, France
- MR Author ID: 632716
- Email: jerome.dedecker@parisdescartes.fr
- F. Merlevède
- Affiliation: Université Gustave Eiffel, Univ Paris Est Créteil, LAMA, UMR 8050 CNRS, 5 Boulevard Descartes, 77454 Marne-la-Vallée, France
- Email: florence.merlevede@univ-eiffel.fr
- Received by editor(s): July 16, 2019
- Received by editor(s) in revised form: March 9, 2020
- Published electronically: October 26, 2020
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 137-174
- MSC (2010): Primary 60F17, 60G42; Secondary 60G50, 22E40
- DOI: https://doi.org/10.1090/tran/8252
- MathSciNet review: 4188180