Convergence to stable laws for a class of multidimensional stochastic recursions

Probability Theory and Related Fields

Volumn 148, Issue 3-4, 2010, Pages 333-402

  Buraczewski, Dariusz ^a   Damek, Ewa ^a   Guivarc'h, Yves ^b  

^a UNIVERSITY OF WROC AW   (Poland)

^b UNIVERSITÉ DE RENNES 1   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77955769683     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-009-0233-7     Document Type: Article

References (31)

1. Aaronson J., Denker M.: Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps. Stoch. Dyn. 1(2), 193-237 (2001).
2. Babillot M., Peigné M.: Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps. Bull. Soc. Math. France 134(1), 119-163 (2006).
3. n with stationary coefficients. Adv. Appl. Probab. 18, 211-220 (1986).
4. Breiman, L.: Probability. Addison-Wesley, Reading (1968).
5. Broise, A., Dal'bo, F., Peigné, M.: Études spectrales d'opérateurs de transfert et applications. Astérisque, 238 (1996).
6. Buraczewski D.: On tails of stationary measures on a class of solvable groups. Ann. Inst. H. Poincaré Probab. Stat. 43(4), 417-440 (2007).
7. Buraczewski, D., Damek, E., Guivarc'h, Y., Hulanicki, A., Urban, R.: Tail-homogeneity of stationary measures for some multidimensional stochastic recursions. Probab. Theory Relat. Fields (to appear).
8. Bougerol, P., Lacroix, J.: Products of random matrices with applications to Schrödinger operators. Progress in Probability and Statistics, vol. 8. Birkhauser, Basel (1985).
9. De Calan C., Luck J., Nienwenhnizen T. M., Petritis D.: On the distribution of a random variable occuring in i. i. d disordered systems. J. Phys. A 18(3), 501-523 (1983).
10. Diaconis P., Freedman D.: Iterated random functions. SIAM Rev. 41(1), 45-76 (1999) (electronic).
11. Feller W.: An introduction to probability theory and its applications II. Wiley, New York (1971).
12. Goldie Ch. M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1(1), 126-166 (1991).
13. Grincevičius A. K.: On limit distribution for a random walk on the line. Lithuanian Math. J. 15, 580-589 (1975).
14. Grincevičius A. K.: Products of random affine transformations. Lithuanian Math. J. 20, 279-282 (1980).
15. Guivarc'h, Y.: Heavy tail properties of multidimensional stochastic recursions. Dynamics and Stochastics: Festschrift in honor of M. S. Keane, vol. 48 of IMS Lecture Notes Monogr. Ser. Inst. Math. Statist., Beachwood, OH, pp. 85-99 (2006).
16. Guivarc'h Y., Hardy J.: Théorèmes limites pour une classe de chaines de Markov et applications aux difféomorphismes d'Anosov. Ann. Inst. H. Poincaré Probab. Stat. 24(1), 73-98 (1988).
17. Guivarc'h Y., Le Jan Y.: symptotic winding of the geodesic flow on modular surfaces and continued fractions. Ann. Sci. Ecole Norm. Sup. (4) 26(1), 23-50 (1993).
18. Guivarc'h Y., Le Page É.: On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks. Ergodic Theory Dyn. Syst. 28, 423-446 (2008).
19. Guivarc'h, Y., Le Page, É.: On the tails of stationary laws for affine stochastic recursions. Preprint, Rennes, 2005.
20. Hennion H.: Sur un théorème spectral et son application aux noyaux lipchitziens. Proc. Am. Math. Soc. 118, 627-634 (1993).
21. Hennion, H., Hervé, L.: Limit Theorems for Markov Chains and Stochastic properties of Dynamical Systems by Quasi Compactness. Lecture Notes in Math. 1766, Springer, Berlin.
22. Hennion H., Hervé L.: Central limit theorems for iterated random Lipschitz mappings. Ann. Probab. 32(3A), 1934-1984 (2004).
23. Ionescu Tulcea, C. T., Marinescu, G.: Théorie ergodique pour des classes d'opérations non complètement continues. Ann. Math. (2) 52, 140-147 (1950).
24. Jurek, Z. J., Mason, J. D.: Operator-limit distributions in probability theory. Wiley, New York, xvi+292 pp (1993).
25. Keller G., Liverani C.: Stability of the spectrum for transfer operators. Ann. Scuola Norm. Sup. Pisa. Cl. Sci. (4) 28, 141-152 (1999).
26. Kesten H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207-248 (1973).
27. Le Page, É.: Théorème de renouvellement pour les produits de matrices aléatoires. Equations aux différences aléatoires. Séminaires de probabilités Rennes 1983, 116 pp, Univ. Rennes I, Rennes (1983).
28. Le Page É.: Régularité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes et applications. Ann. Inst. H. Poincaré Probab. Stat. 25(2), 109-142 (1989).
29. Lévy P.: Théorie de l'Addition des Variables Aléatoires. Gauthier-Villars, Paris (1954).
30. Sarig O.: Continuous phase transitions for dynamical systems. Comm. Math. Phys. 267(3), 631-667 (2006).
31. Varopoulos N. Th.: Wiener-Hopf theory and nonunimodular groups. J. Funct. Anal. 120(2), 467-483 (1994).