Central limit theorems in linear dynamics
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 1131-1158.

Étant donné un opérateur T agissant sur un espace de Banach X, nous étudions l’existence d’une mesure de probabilité μ sur X telle que, pour de nombreuses fonctions f:X𝕂, la suite (f++fT n-1 )/n converge en loi vers une variable aléatoire gaussienne.

Given a bounded operator T on a Banach space X, we study the existence of a probability measure μ on X such that, for many functions f:X𝕂, the sequence (f++fT n-1 )/n converges in distribution to a Gaussian random variable.

DOI : 10.1214/13-AIHP585
Mots clés : hypercyclic operators, linear dynamics, ergodic theory, dynamical systems, central limit theorem
@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  - 
%0 Journal Article
%A Bayart, Frédéric
%T Central limit theorems in linear dynamics
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2015
%P 1131-1158
%V 51
%N 3
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/13-AIHP585/
%R 10.1214/13-AIHP585
%G en
%F AIHPB_2015__51_3_1131_0
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] F. Bayart and S. Grivaux. Frequently hypercyclic operators. Trans. Amer. Math. Soc. 358 (11) (2006) 5083–5117. | MR | Zbl

[2] F. Bayart and S. Grivaux. Invariant Gaussian measures for operators on Banach spaces and linear dynamics. Proc. Lond. Math. Soc. 94 (2007) 181–210. | DOI | MR | Zbl

[3] F. Bayart and É. Matheron. Dynamics of Linear Operators. Cambridge Tracts in Math. 179. Cambridge Univ. Press, Cambridge, 2009. | DOI | MR | Zbl

[4] F. Bayart and É. Matheron. Mixing operators and small subsets of the circle. J. Reine Angew. Math. To appear, 2015. Available at arXiv:1112.1289.

[5] F. Bayart and I. Ruzsa. Difference sets and frequently hypercyclic weighted shifts. Ergodic Theory Dynam. Syst. 35 (2015) 691–709. | DOI | MR | Zbl

[6] S. Cantat and S. Le Borgne. Théorème central limite pour les endomorphismes holomorphes et les correspondances modulaires. Int. Math. Res. Not. 56 (2005) 3479–3510. | DOI | MR | Zbl

[7] N. I. Chernov. Limit theorems and Markov approximations for chaotic dynamical systems. Probab. Theory Related Fields 101 (1995) 321–362. | DOI | MR | Zbl

[8] V. Devinck. Strongly mixing operators on Hilbert spaces and speed of mixing. Proc. Lond. Math. Soc. 106 (2013) 1394–1434. | DOI | MR | Zbl

[9] T. C. Dinh and N. Sibony. Decay of correlations and central limit theorem for meromorphic maps. Comm. Pure Appl. Math. 59 (2006) 754–768. | DOI | MR | Zbl

[10] C. Dupont. Bernoulli coding map and almost-sure invariance principle for endomorphisms of k . Probab. Theory Related Fields 146 (2010) 337–359. | DOI | MR | Zbl

[11] E. Flytzanis. Unimodular eigenvalues and linear chaos in Hilbert spaces. Geom. Funct. Anal. 5 (1995) 1–13. | DOI | MR | Zbl

[12] K.-G. Grosse-Erdmann and A. Peris. Linear Chaos. Springer, Berlin, 2011. | DOI | MR | Zbl

[13] M. I. Gordin. The central limit theorem for stationary processes. Soviet Math. Dokl. 10 (1969) 1174–1176. | MR | Zbl

[14] S. Le Borgne. Limit theorems for non-hyperbolic automorphisms of the torus. Israel J. Math. 109 (1999) 61–73. | DOI | MR | Zbl

[15] C. Liverani. 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] M. Murillo-Arcila and A. Peris. Strong mixing measures for linear operators and frequent hypercyclicity. J. Math. Anal. Appl. 398 (2013) 462–465. | DOI | MR | Zbl

[17] M. Maxwell and M. Woodroofe. Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 (2000) 713–724. | DOI | MR | Zbl

[18] R. Rudnicki. Strong ergodic properties of a first-order partial differential equation. J. Math. Anal. Appl. 133 (1988) 14–26. | DOI | MR | Zbl

[19] R. Rudnicki. Gaussian measure-preserving linear transformations. Univ. Iagel. Acta Math. 30 (1993) 105–112. | MR | Zbl

[20] R. Rudnicki. Chaos for some infinite-dimensional dynamical systems. Math. Methods Appl. Sci. 27 (2004) 723–738. | DOI | MR | Zbl

[21] D. Volný. On limit theorems and category for dynamical systems. Yokohama Math. J. 38 (1990) 29–35. | MR | Zbl

[22] D. Volný. 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

Cité par Sources :