Stable laws and spectral gap properties for affine random walks

Gao, Zhiqiang; Guivarc’h, Yves; Le Page, Émile

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 319-348.

Nous considérons une relation de récurrence affine multidimensionelle à coefficients aléatoires et nous supposons que l’opérateur de Markov $P$ associé a une unique probabilité stationnaire. Nous montrons la propriété de trou spectral pour les opérateurs de Fourier correspondants sur certains espaces de fonctions Holdériennes, et nous en déduisons la convergence vers des lois stables pour les sommes de Birkhoff le long des trajectoires. Les paramètres des lois stables obtenues s’expriment à l’aide de quantités dépendant essentiellement de la partie multiplicative de $P$. La preuve est basée sur les propriétés spectrales de l’opérateur de Markov associé et l’homogénéité à l’infini de la mesure stationnaire.

We consider a general multidimensional affine recursion with corresponding Markov operator $P$ and a unique $P$-stationary measure. We show spectral gap properties on Hölder spaces for the corresponding Fourier operators and we deduce convergence to stable laws for the Birkhoff sums along the recursion. The parameters of the stable laws are expressed in terms of basic quantities depending essentially on the matricial multiplicative part of $P$. Spectral gap properties of $P$ and homogeneity at infinity of the $P$-stationary measure play an important role in the proofs.

[1] G. Alsmeyer and S. Mentemeier. Tail behaviour of stationary solutions of random difference equations: The case of regular matrices. J. Difference Equ. Appl. 18 (2012) 1305–1332. | DOI | MR | Zbl

[2] M. Babillot and M. Peigné. Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps. Bull. Soc. Math. France 134 (2006) 119–163. | DOI | Numdam | MR | Zbl

[3] P. Bougerol and J. Lacroix. Products of Random Matrices with Applications to Schrödinger Operators. Progress in Probability and Statistics 8. Birkhäuser Boston, Boston, MA, 1985. | DOI | MR | Zbl

[4] P. Bougerol and N. Picard. Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 (1992) 1714–1730. | DOI | MR | Zbl

[5] D. Buraczewski, E. Damek and Y. Guivarc’H. Convergence to stable laws for a class of multidimensional stochastic recursions. Probab. Theory Related Fields 148 (2010) 333–402. | DOI | MR | Zbl

[6] D. Buraczewski, E. Damek, Y. Guivarc’H, A. Hulanicki and R. Urban. Tail-homogeneity of stationary measures for some multidimensional stochastic recursions. Probab. Theory Related Fields 145 (2009) 385–420. | DOI | MR | Zbl

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

[8] J.-P. Conze and Y. Guivarc’H. Ergodicity of group actions and spectral gaps, applications to random walks and Markov shifts. Discrete Contin. Dyn. Syst. 33 (2013) 4239–4269. | DOI | MR | Zbl

[9] E. Damek, S. Mentemeier, M. Mirek and J. Zienkiewicz. Convergence to stable laws for multidimensional stochastic recursions: The case of regular matrices. Potential Anal. 38 (2013) 683–697. | DOI | MR | Zbl

[10] D. Dolgopyat. On mixing properties of compact group extensions of hyperbolic systems. Israel J. Math. 130 (2002) 157–205. | DOI | MR | Zbl

[11] A. Furman and Y. Shalom. Sharp ergodic theorems for group actions and strong ergodicity. Ergodic Theory Dynam. Systems 19 (1999) 1037–1061. | DOI | MR | Zbl

[12] H. Furstenberg. Boundary theory and stochastic processes on homogeneous spaces. In Harmonic Analysis on Homogeneous Spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) 193–229. Amer. Math. Soc., Providence, RI, 1973. | MR | Zbl

[13] C. M. Goldie. Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 (1991) 126–166. | DOI | MR | Zbl

[14] I. Y. Goldsheid. Linear and sub-linear growth and the CLT for hitting times of a random walk in random environment on a strip. Probab. Theory Related Fields 141 (2008) 471–511. | DOI | MR | Zbl

[15] I. Y. Goldsheid and Y. Guivarc’H. Zariski closure and the dimension of the Gaussian law of the product of random matrices. I. Probab. Theory Related Fields 105 (1996) 109–142. | DOI | MR | Zbl

[16] S. Gouëzel. Local limit theorem for nonuniformly partially hyperbolic skew-products and Farey sequences. Duke Math. J. 147 (2009) 193–284. | DOI | MR | Zbl

[17] Y. Guivarc’H. Heavy tail properties of stationary solutions of multidimensional stochastic recursions. In Dynamics & Stochastics 85–99. IMS Lecture Notes Monogr. Ser. 48. Inst. Math. Statist., Beachwood, OH, 2006. | MR | Zbl

[18] Y. Guivarc’H. On contraction properties for products of Markov driven random matrices. Zh. Mat. Fiz. Anal. Geom. 4 (2008) 457–489, 573. | MR | Zbl

[19] Y. Guivarc’H and J. Hardy. Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. Ann. Inst. Henri Poincaré Probab. Stat. 24 (1988) 73–98. | Numdam | MR | Zbl

[20] Y. Guivarc’H and É. Le Page. Simplicité de spectres de Lyapounov et propriété d’isolation spectrale pour une famille d’opérateurs de transfert sur l’espace projectif. In Random Walks and Geometry 181–259. Walter de Gruyter, Berlin, 2004. | MR | Zbl

[21] Y. Guivarc’H and E. Le Page. On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks. Ergodic Theory Dynam. Systems 28 (2008) 423–446. | DOI | MR | Zbl

[22] Y. Guivarc’H and E. Le Page. Spectral gap properties and asymptotics of stationary measures for affine random walks, 2012. Available at arXiv:1204.6004v1.

[23] Y. Guivarc’H and A. Raugi. Products of random matrices: Convergence theorems. In Random Matrices and Their Applications (Brunswick, Maine, 1984) 31–54. Contemp. Math. 50. Amer. Math. Soc., Providence, RI, 1986. | MR | Zbl

[24] Y. Guivarc’H and R. Urban. Semigroup actions on tori and stationary measures on projective spaces. Studia Math. 171 (2005) 33–66. | DOI | MR | Zbl

[25] H. Hennion and L. Hervé. Central limit theorems for iterated random Lipschitz mappings. Ann. Probab. 32 (2004) 1934–1984. | DOI | MR | Zbl

[26] W. Hong and H. Wang. (2010). Branching structure for an (L-1) random walk in random environment and its applications. Available at arXiv:1003.3731v1. | DOI | MR | Zbl

[27] I. A. Ibragimov and Y. V. Linnik. Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov, Translation from the Russian edited by J. F. C. Kingman. | MR | Zbl

[28] C. T. Ionescu Tulcea and G. Marinescu. Théorie ergodique pour des classes d’opérations non complètement continues. Ann. of Math. (2) 52 (1950) 140–147. | MR | Zbl

[29] Z. J. Jurek and J. D. Mason. Operator-Limit Distributions in Probability Theory. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York, 1993. | MR | Zbl

[30] G. Keller and C. Liverani. Stability of the spectrum for transfer operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999) 141–152. | Numdam | MR | Zbl

[31] H. Kesten. Random difference equations and renewal theory for products of random matrices. Acta Math. 131 (1973) 207–248. | DOI | MR | Zbl

[32] H. Kesten. Renewal theory for functionals of a Markov chain with general state space. Ann. Probab. 2 (1974) 355–386. | DOI | MR | Zbl

[33] H. Kesten, M. V. Kozlov and F. Spitzer. A limit law for random walk in a random environment. Compos. Math. 30 (1975) 145–168. | Numdam | MR | Zbl

[34] É. Le Page. Théorèmes de renouvellement pour les produits de matrices aléatoires. Équations aux différences aléatoires. In Séminaires de probabilités Rennes. Publ. Sém. Math. 116. Univ. Rennes I, Rennes, 1983. | MR

[35] É. Le Page. Régularité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes et applications. Ann. Inst. Henri Poincaré Probab. Stat. 25 (1989) 109–142. | Numdam | MR | Zbl

[36] P. Lévy. Théorie de l’Addition des Variables Aléatoires, 2th edition. Monographies des probabilités; calcul des probabilités et ses applications, Fasc. 1. Gauthier-Villars, Paris, FR, 1954. | JFM | Zbl

[37] T. Mikosch and C. Stărică. Limit theory for the sample autocorrelations and extremes of a $GARCH(1,1)$ process. Ann. Statist. 28 (2000) 1427–1451. | DOI | MR | Zbl

[38] M. Mirek. Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps. Probab. Theory Related Fields 151 (2011) 705–734. | DOI | MR | Zbl

[39] G. Prasad. $𝐑$-regular elements in Zariski-dense subgroups. Quart. J. Math. Oxford Ser. (2) 45 (1994) 541–545. | MR | Zbl

[40] G. Samorodnitsky and M. S. Taqqu. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Stochastic Modeling. Chapman & Hall, New York, 1994. | MR | Zbl

[41] F. Solomon. Random walks in a random environment. Ann. Probab. 3 (1975) 1–31. | DOI | MR | Zbl