Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-24T16:13:12.026Z Has data issue: false hasContentIssue false
  • English
  • Français

On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks

Published online by Cambridge University Press:  01 April 2008

Y. GUIVARC’H  and
EMILE LE PAGE
Y. GUIVARC’H
Affiliation:
IRMAR, CNRS Rennes I, Université de Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France (email: yves.guivarch@univ-rennes1.fr)
EMILE LE PAGE
Affiliation:
LMAM, Université de Bretagne Sud, Campus de Tohannic, BP 573, 56000 Vannes-Cedex, France (email: lepage@univ-ubs.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a random walk on the affine group of the real line, we denote by P the corresponding Markov operator on , and we study the Birkhoff sums associated with its trajectories. We show that, depending on the parameters of the random walk, the normalized Birkhoff sums converge in law to a stable law of exponent α∈ ]0,2[ or to a normal law. The corresponding analysis is based on the spectral properties of two families of associated transfer operators Pt,Tt. The operator Pt is a Fourier operator and is considered here as a perturbation of the Markov operator P=P0 of the random walk. The operator Tt is related to Pt by a symmetry of Heisenberg type and is also considered as a perturbation of the Markov operator T0=T. We prove that these operators have an isolated dominant eigenvalue which has an asymptotic expansion involving fractional powers of t. The parameters of this expansion have simple expressions in terms of tails and moments of the stationary measures of P and T.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 2: William Parry Memorial Volume , April 2008 , pp. 423 - 446
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aaronson, J. and Denker, M.. Local limit theorems for partial sums of stationary sequences generated by Gibbs–Markov maps. Stoch. Dyn. 1(2) (2001), 193–237.CrossRefGoogle Scholar
[2]Babillot, M. and Peigné, M.. Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps. Bull. Soc. Math. France 134(1) (2006), 119–163.Google Scholar
[3]Bolthausen, E. and Goldsheid, I.. Recurrence and transience of random walks in random environnements in a strip. Com. Math. Phys. 214 (2000), 429–447.Google Scholar
[4]Bougerol, P. and Lacroix, J.. Products of Random Matrices with Applications to Schröedinger Operators (Progress in Probability and Statistics, 8). Birkhäuser, Boston, 1985.CrossRefGoogle Scholar
[5]Bremont, J.. On some random walks on , in random medium. Ann. Probab. 30 (2002), 1266–1312.Google Scholar
[6]Broise, A., Dal’bo, F. and Peigné, M.. Etudes spectrales d’opérateurs de transfert et applications (Astérisque, 238). Société Mathématique de France, Paris, 1996.Google Scholar
[7]Derrida, B. and Hilhorst, H.. Singular behaviour of certain infinite products of random 2×2 matrices. J. Phys. A. Math. Gen. 16 (1983), 2641–2654.CrossRefGoogle Scholar
[8]Doeblin, W. and Fortet, R.. Sur les chaǐnes à liaisons complètes. Bull. Soc. Math. France 65 (1937), 132–148.CrossRefGoogle Scholar
[9]Gnedenko, B. V. and Kolmogorov, A. N.. Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, MA, 1954.Google Scholar
[10]Goldie, C. M.. Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 4(1) (1991), 126–166.Google Scholar
[11]Gouezel, S.. Central limit theorem and stable laws for intermittent maps. Probab. Theory Related Fields 128(1) (2004), 82–122.CrossRefGoogle Scholar
[12]Grenander, U.. Probabilities on Algebraic Structures. Wiley, New York, 1963.Google Scholar
[13]Guivarc’h, Y. and Hardy, J.. Théorèmes limites pour une classe de chaǐnes de Markov et applications aux difféomorphismes d’Anosov. Ann. Inst. H. Poincaré 24 (1988), 73–98.Google Scholar
[14]Guivarc’h, Y.. Heavy tail properties of multidimensional stochastic recursions. Dyn. Stoch. 48 (2006), 85–99.Google Scholar
[15]Guivarc’h, Y. and Le Jan, Y.. Asymptotic winding of the geodesic flow on modular surfaces and continuous fractions. Ann. Sci. École Norm. Sup. (4) 26 (1993), 23–50.Google Scholar
[16]Guivarc’h, Y. and Le Page, E.. On the tails of stationary laws for stochastic recursions. Preprint Rennes, 2005.Google Scholar
[17]Hennion, H. and Hervé, L.. Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness (Lecture Notes in Mathematics, 1766). Springer, Berlin, 2001.CrossRefGoogle Scholar
[18]Keller, G.. Un théorème central limite pour une classe de transformations monotones par morceaux. C. R. Acad. Sci., Paris 291 (1980), 155–158.Google Scholar
[19]Keller, G. and Liverani, C.. Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa 28(1) (1999), 141–152.Google Scholar
[20]Kesten, A., Kozlov, M. V. and Spitzer, F.. A limit law for a random walk in a random environment. Compos. Math. 30 (1975), 145–168.Google Scholar
[21]Le Page, E.. Regularité du plus grand exposant caractéristique des produits de matrices aléatoires et applications. Ann. Inst. H. Poincaré B 25(2) (1989), 109–142.Google Scholar
[22]Nagaev, S. V.. Some limit theorems for stationary Markov chains. Theory Probab. Appl. 11(4) (1957), 378–406.Google Scholar
[23]Parry, W. and Pollicott, M.. Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics (Asterisque). Société Mathématique de France, Paris, 1990, pp. 187–188.Google Scholar
[24]Rousseau Egele, J.. Un théorème de la limite locale pour une classe de transformations dilatantes. Ann. Probab. 11(3) (1983), 772–788.CrossRefGoogle Scholar
[25]Solomon, F.. Random walk in a random environment. Ann. Probab. 8(1) (1975), 1–31.Google Scholar