Weak convergence to stable Lévy processes for nonuniformly hyperbolic dynamical systems

Melbourne, Ian; Zweimüller, Roland

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 545-556.

Nous considérons des principes d’invariance faibles (théorèmes limites fonctionnels) dans le domaine d’une loi stable. Un résultat général est obtenu en relevant de telles lois limites depuis un système dynamique induit vers le système original. Une classe importante d’exemples couverte par notre résultat est donnée par les transformations intermittentes à la Pomeau–Manneville, où la convergence pour le système induit est dans la topologie ${𝒥}_1$ de Skorohod standard. Pour le système complet, il n’y a pas de convergence dans la topologie ${𝒥}_1$, mais nous prouvons la convergence dans la topologie ${ℳ}_1$.

We consider weak invariance principles (functional limit theorems) in the domain of a stable law. A general result is obtained on lifting such limit laws from an induced dynamical system to the original system. An important class of examples covered by our result are Pomeau–Manneville intermittency maps, where convergence for the induced system is in the standard Skorohod ${𝒥}_1$ topology. For the full system, convergence in the ${𝒥}_1$ topology fails, but we prove convergence in the ${ℳ}_1$ topology.

[1] J. Aaronson and M. Denker. Local limit theorems for partial sums of stationary sequences generated by Gibbs–Markov maps. Stoch. Dyn. 1 (2001) 193–237. | DOI | MR | Zbl

[2] F. Avram and M. S. Taqqu. Weak convergence of sums of moving averages in the $\alpha$-stable domain of attraction. Ann. Probab. 20 (1992) 483–503. | DOI | MR | Zbl

[3] P. Bálint, N. Chernov and D. Dolgopyat. Limit theorems for dispersing billiards with cusps. Comm. Math. Phys. 308 (2011) 479–510. | DOI | MR | Zbl

[4] P. Bálint and S. Gouëzel. Limit theorems in the stadium billiard. Comm. Math. Phys. 263 (2006) 461–512. | DOI | MR | Zbl

[5] P. Bálint and I. Melbourne. Decay of correlations and invariance principles for dispersing billiards with cusps, and related planar billiard flows. J. Stat. Phys. 133 (2008) 435–447. | DOI | MR | Zbl

[6] G. Ben Arous and J. Černý. Scaling limit for trap models on ${\mathbb{Z}}^d$. Ann. Probab. 35 (2007) 2356–2384. | DOI | MR | Zbl

[7] M. Benedicks and L.-S. Young. Markov extensions and decay of correlations for certain Hénon maps. Astérisque 261 (2000) 13–56. | Numdam | MR | Zbl

[8] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York, 1999. | DOI | MR | Zbl

[9] D. L. Burkholder. Distribution function inequalities for martingales. Ann. Probab. 1 (1973) 19–42. | DOI | MR | Zbl

[10] J.-P. Conze and S. Le Borgne. Méthode de martingales et flow géodésique sur une surface de courbure constante négative. Ergodic Theory Dynam. Systems 21 (2001) 421–441. | DOI | MR | Zbl

[11] J. Dedecker and F. Merlevède. Weak invariance principle and exponential bounds for some special functions of intermittent maps. High Dimensional Probability 5 (2009) 60–72. | MR | Zbl

[12] M. Denker and W. Philipp. Approximation by Brownian motion for Gibbs measures and flows under a function. Ergodic Theory Dynam. Systems 4 (1984) 541–552. | DOI | MR | Zbl

[13] J. L. Doob. Stochastic Processes. Wiley, New York, 1953. | MR | Zbl

[14] G. K. Eagleson. Some simple conditions for limit theorems to be mixing. Teor. Verojatnost. i Primenen 21 (1976) 653–660. | MR | Zbl

[15] M. J. Field, I. Melbourne and A. Török. Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions. Ergodic Theory Dynam. Systems 23 (2003) 87–110. | DOI | MR | Zbl

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

[17] G. A. Gottwald and I. Melbourne. Central limit theorems and suppression of anomalous diffusion for systems with symmetry. Preprint, 2012.

[18] S. Gouëzel. Central limit theorem and stable laws for intermittent maps. Probab. Theory Related Fields 128 (2004) 82–122. | DOI | MR | Zbl

[19] S. Gouëzel. Statistical properties of a skew product with a curve of neutral points. Ergodic Theory Dynam. Systems 27 (2007) 123–151. | DOI | MR | Zbl

[20] S. Gouëzel. Almost sure invariance principle for dynamical systems by spectral methods. Ann. Probab. 38 (2010) 1639–1671. | DOI | MR | Zbl

[21] F. Hofbauer and G. Keller. Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 (1982) 119–140. | DOI | MR | Zbl

[22] M. Holland and I. Melbourne. Central limit theorems and invariance principles for Lorenz attractors. J. London Math. Soc. 76 (2007) 345–364. | DOI | MR | Zbl

[23] C. Liverani, B. Saussol and S. Vaienti. A probabilistic approach to intermittency. Ergodic Theory Dynam. Systems 19 (1999) 671–685. | DOI | MR | Zbl

[24] I. Melbourne and M. Nicol. Statistical properties of endomorphisms and compact group extensions. J. London Math. Soc. 70 (2004) 427–446. | DOI | MR | Zbl

[25] I. Melbourne and M. Nicol. Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260 (2005) 131–146. | DOI | MR | Zbl

[26] I. Melbourne and M. Nicol. A vector-valued almost sure invariance principle for hyperbolic dynamical systems. Ann. Probab. 37 (2009) 478–505. | DOI | MR | Zbl

[27] I. Melbourne and A. Török. Central limit theorems and invariance principles for time-one maps of hyperbolic flows. Comm. Math. Phys. 229 (2002) 57–71. | DOI | MR | Zbl

[28] I. Melbourne and A. Török. Statistical limit theorems for suspension flows. Israel J. Math. 144 (2004) 191–209. | DOI | MR | Zbl

[29] Y. Pomeau and P. Manneville. Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 74 (1980) 189–197. | DOI | MR | Zbl

[30] M. Ratner. The central limit theorem for geodesic flows on $n$-dimensional manifolds of negative curvature. Israel J. Math. 16 (1973) 181–197. | DOI | MR | Zbl

[31] A. V. Skorohod. Limit theorems for stochastic processes. Teor. Veroyatnost. i Primenen. 1 (1956) 289–319. | MR | Zbl

[32] D. Szász and T. Varjú. Limit laws and recurrence for the planar Lorentz process with infinite horizon. J. Stat. Phys. 129 (2007) 59–80. | DOI | MR | Zbl

[33] M. Tyran-Kamińska. Weak convergence to Lévy stable processes in dynamical systems. Stoch. Dyn. 10 (2010) 263–289. | DOI | MR | Zbl

[34] W. Whitt. Stochastic-Process Limits. Springer Series in Operations Research. Springer, New York, 2002. | DOI | MR | Zbl

[35] L.-S. Young. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. 147 (1998) 585–650. | DOI | MR | Zbl

[36] L.-S. Young. Recurrence times and rates of mixing. Israel J. Math. 110 (1999) 153–188. | DOI | MR | Zbl

[37] R. Zweimüller. Stable limits for probability preserving maps with indifferent fixed points. Stoch. Dyn. 3 (2003) 83–99. | DOI | MR | Zbl

[38] R. Zweimüller. Mixing limit theorems for ergodic transformations. J. Theoret. Probab. 20 (2007) 1059–1071. | DOI | MR | Zbl