Central limit theorems in linear dynamics

Bayart, Frédéric

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 $\mu$ sur $X$ telle que, pour de nombreuses fonctions $f:X\to 𝕂$, la suite $(f+\cdots +f\circ T^{n-1})/\sqrt{n}$ 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 $\mu$ on $X$ such that, for many functions $f:X\to 𝕂$, the sequence $(f+\cdots +f\circ T^{n-1})/\sqrt{n}$ converges in distribution to a Gaussian random variable.

Mots clés : hypercyclic operators, linear dynamics, ergodic theory, dynamical systems, central limit theorem

[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 ${\mathbb{P}}^k$. 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