Étant donné un opérateur agissant sur un espace de Banach , nous étudions l’existence d’une mesure de probabilité sur telle que, pour de nombreuses fonctions , la suite converge en loi vers une variable aléatoire gaussienne.
Given a bounded operator on a Banach space , we study the existence of a probability measure on such that, for many functions , the sequence converges in distribution to a Gaussian random variable.
@article{AIHPB_2015__51_3_1131_0, author = {Bayart, Fr\'ed\'eric}, title = {Central limit theorems in linear dynamics}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {1131--1158}, publisher = {Gauthier-Villars}, volume = {51}, number = {3}, year = {2015}, doi = {10.1214/13-AIHP585}, mrnumber = {3365976}, zbl = {1353.47015}, language = {en}, url = {http://www.numdam.org/articles/10.1214/13-AIHP585/} }
TY - JOUR AU - Bayart, Frédéric TI - Central limit theorems in linear dynamics JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 1131 EP - 1158 VL - 51 IS - 3 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/13-AIHP585/ DO - 10.1214/13-AIHP585 LA - en ID - AIHPB_2015__51_3_1131_0 ER -
Bayart, Frédéric. Central limit theorems in linear dynamics. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 1131-1158. doi : 10.1214/13-AIHP585. http://www.numdam.org/articles/10.1214/13-AIHP585/
[1] Frequently hypercyclic operators. Trans. Amer. Math. Soc. 358 (11) (2006) 5083–5117. | MR | Zbl
and .[2] Invariant Gaussian measures for operators on Banach spaces and linear dynamics. Proc. Lond. Math. Soc. 94 (2007) 181–210. | DOI | MR | Zbl
and .[3] Dynamics of Linear Operators. Cambridge Tracts in Math. 179. Cambridge Univ. Press, Cambridge, 2009. | DOI | MR | Zbl
and .[4] Mixing operators and small subsets of the circle. J. Reine Angew. Math. To appear, 2015. Available at arXiv:1112.1289.
and .[5] Difference sets and frequently hypercyclic weighted shifts. Ergodic Theory Dynam. Syst. 35 (2015) 691–709. | DOI | MR | Zbl
and .[6] Théorème central limite pour les endomorphismes holomorphes et les correspondances modulaires. Int. Math. Res. Not. 56 (2005) 3479–3510. | DOI | MR | Zbl
and .[7] Limit theorems and Markov approximations for chaotic dynamical systems. Probab. Theory Related Fields 101 (1995) 321–362. | DOI | MR | Zbl
.[8] Strongly mixing operators on Hilbert spaces and speed of mixing. Proc. Lond. Math. Soc. 106 (2013) 1394–1434. | DOI | MR | Zbl
.[9] Decay of correlations and central limit theorem for meromorphic maps. Comm. Pure Appl. Math. 59 (2006) 754–768. | DOI | MR | Zbl
and .[10] Bernoulli coding map and almost-sure invariance principle for endomorphisms of . Probab. Theory Related Fields 146 (2010) 337–359. | DOI | MR | Zbl
.[11] Unimodular eigenvalues and linear chaos in Hilbert spaces. Geom. Funct. Anal. 5 (1995) 1–13. | DOI | MR | Zbl
.[12] Linear Chaos. Springer, Berlin, 2011. | DOI | MR | Zbl
and .[13] The central limit theorem for stationary processes. Soviet Math. Dokl. 10 (1969) 1174–1176. | MR | Zbl
.[14] Limit theorems for non-hyperbolic automorphisms of the torus. Israel J. Math. 109 (1999) 61–73. | DOI | MR | Zbl
.[15] Central limit theorem for deterministic systems. In International Conference on Dynamical Systems (Montevideo, 1995). Pitman Res. Notes Math. Ser. 362 56–75. Longman, Harlow, 1996. | MR | Zbl
.[16] Strong mixing measures for linear operators and frequent hypercyclicity. J. Math. Anal. Appl. 398 (2013) 462–465. | DOI | MR | Zbl
and .[17] Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 (2000) 713–724. | DOI | MR | Zbl
and .[18] Strong ergodic properties of a first-order partial differential equation. J. Math. Anal. Appl. 133 (1988) 14–26. | DOI | MR | Zbl
.[19] Gaussian measure-preserving linear transformations. Univ. Iagel. Acta Math. 30 (1993) 105–112. | MR | Zbl
.[20] Chaos for some infinite-dimensional dynamical systems. Math. Methods Appl. Sci. 27 (2004) 723–738. | DOI | MR | Zbl
.[21] On limit theorems and category for dynamical systems. Yokohama Math. J. 38 (1990) 29–35. | MR | Zbl
.[22] Martingale approximation of non adapted stochastic processes with nonlinear growth of variance. In Dependence in Probability and Statistics. Lecture Notes in Statistics 187 141–156. Springer, Berlin, 2006. | MR | Zbl
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