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Exponential chi-squared distributions in infinite ergodic theory

Published online by Cambridge University Press:  01 February 2013

JON AARONSON  and
OMRI SARIG
JON AARONSON
Affiliation:
School of Math. Sciences, Tel Aviv University, 69978 Tel Aviv, Israel (email: aaro@tau.ac.il)
OMRI SARIG
Affiliation:
Faculty of Mathematics and Computer Sciences, The Weizmann Institute for Science, POB 26, Rehovot 76100, Israel (email: omsarig@gmail.com)
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Abstract

We prove distributional limit theorems for random walk adic transformations obtaining ergodic distributional limits of exponential $\chi ^2$form.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 3 , June 2014 , pp. 705 - 724
Copyright
©2013 Cambridge University Press 

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References

[A1]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.Google Scholar
[A2]Aaronson, J.. The asymptotic distributional behaviour of transformations preserving infinite measures. J. Anal. Math. 39 (1981), 203–234.Google Scholar
[AD]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
[ADSZ]Aaronson, J., Denker, M., Sarig, O. and Zweimüller, R.. Aperiodicity of cocycles and conditional local limit theorems. Stoch. Dyn. 4(1) (2004), 31–62.Google Scholar
[ANSS]Aaronson, J., Nakada, H., Sarig, O. and Solomyak, R.. Invariant measures and asymptotics for some skew products. Israel J. Math. 128 (2002), 93–134. Corrections: Israel J. Math. 138 (2003), 377–379.Google Scholar
[ANS]Aaronson, J., Nakada, H. and Sarig, O.. Exchangeable measures for subshifts. Ann. Inst. Henri Poincaré Probab. Stat. 42(6) (2006), 727–751.CrossRefGoogle Scholar
[AW]Aaronson, J. and Weiss, B.. On the asymptotics of a 1-parameter family of infinite measure preserving transformations. Bol. Soc. Brasil. Mat. (N.S.) 29 (1998), 181–193.CrossRefGoogle Scholar
[BL]Babillot, M. and Ledrappier, F.. Geodesic Paths and Horocycle Flow on Abelian Covers. Lie Groups and Ergodic Theory (Mumbai, 1996) (Tata Institute of Fundamental Research Studies in Mathematics, 14). Tata Institute of Fundamental Research, Bombay, 1998, pp. 1–32.Google Scholar
[BM]Bowen, R. and Marcus, B.. Unique ergodicity for horocycle foliations. Israel J. Math. 26(1) (1977), 43–67.Google Scholar
[DF]Doeblin, W. and Fortet, R.. Sur des chaînes a liaison complètes. Bull. Soc. Math. France 65 (1937), 132–148.CrossRefGoogle Scholar
[DK]Darling, D. A. and Kac, M.. On occupation times for Markoff processes. Trans. Amer. Math. Soc. 84 (1957), 444–458.Google Scholar
[E]Eagleson, G. K.. Some simple conditions for limit theorems to be mixing. Teor. Veroyatn. Primen. 21(3) (1976), 653–660 (in Russian); Engl. Transl. Theory Probab. Appl. 21(3) (1977), 637–642.Google Scholar
[G]Guivarc’h, Y.. Propriétés ergodiques, en mesure infinie, de certains systèmes dynamiques fibrés. Ergod. Th. & Dynam. Sys. 9(3) (1989), 433–453.Google Scholar
[GH]Guivarc’h, Y. and Hardy, J.. Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. Ann. Inst. Henri Poincaré Probab. Stat. 24(1) (1988), 73–98.Google Scholar
[HH]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
[HIK]Hajian, A., Ito, Y. and Kakutani, S.. Invariant measures and orbits of dissipative transformations. Adv. Math. 9 (1972), 52–65.CrossRefGoogle Scholar
[ITM]Ionescu-Tulcea, C. and Marinescu, G.. Théorie ergodique pour des classes d’opérations non complètement continues. Ann. of Math. (2) 47 (1950), 140–147.CrossRefGoogle Scholar
[KS]Katsuda, A. and Sunada, T.. Closed orbits in homology classes. Publ. Math. Inst. Hautes Études Sci. 71 (1990), 5–32.Google Scholar
[LS]Ledrappier, F. and Sarig, O.. Unique ergodicity for non-uniquely ergodic horocycle flows. Discrete Contin. Dyn. Syst. 16(2) (2006), 411–433.Google Scholar
[LM]Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.Google Scholar
[L1]Livšic, A. N.. On the spectra of adic transformations of Markov compact sets. Uspekhi Mat. Nauk 42(3(255)) (1987), 189–190 (in Russian); Engl. Transl. Russian Math. Surveys 42(3) (1987), 222–223.Google Scholar
[L2]Livšic, A. N.. Certain properties of the homology of $Y$-systems. Mat. Zametki 10 (1971), 555–564 (in Russian); Engl. Transl. Math. Notes 10 (1971), 758–763.Google Scholar
[Mrc]Marcus, B.. Unique ergodicity of the horocycle flow: variable curvature case. Israel J. Math. 21 (1975), 133–144.Google Scholar
[Mrg]Margulis, G. A.. Certain measures that are connected with $U$-flows on compact manifolds. Funktsional. Anal. i Prilozhen 4(1) (1970), 62–76 (in Russian).Google Scholar
[N]Nagaev, S. V.. Some limit theorems for stationary Markov chains. Theory Probab. Appl. 2 (1957), 378–406.Google Scholar
[PP]Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990), 268.Google Scholar
[PS]Parry, W. and Schmidt, K.. Natural coefficients and invariants for Markov-shifts. Invent. Math. 76 (1984), 15–32.Google Scholar
[Po]Pollicott, M.. $\mathbb Z^d$-covers of horosphere foliations. Discrete Contin. Dyn. Syst. 6(1) (2000), 147–154.CrossRefGoogle Scholar
[R]Ratner, M.. The central limit theorem for geodesic flows on $n$-dimensional manifolds of negative curvature. Israel J. Math. 16 (1973), 181–197.Google Scholar
[RE]Rousseau-Egele, J.. Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux. Ann. Probab. 11(3) (1983), 772–788.Google Scholar
[Sa]Sarig, O.. Invariant Radon measures for horocycle flows on abelian covers. Invent. Math. 157(3) (2004), 519–551.Google Scholar
[S]Stone, C.. On local and ratio limit theorems. Proc. Fifth Berkeley Symp. on Mathematical Statistics and Probability (Berkeley, CA, 1965/66), Vol. II: Contributions to Probability Theory, Part 2. University of California Press, Berkeley, CA, 1967, pp. 217–224.Google Scholar
[TZ]Thaler, M. and Zweimüller, R.. Distributional limit theorems in infinite ergodic theory. Probab. Theory Related Fields 135 (2006), 15–52.Google Scholar
[V]Vershik, A. M.. A new model of the ergodic transformations. Dynamical Systems and Ergodic Theory (Warsaw, 1986) (Banach Center Publications, 23). PWN, Warsaw, 1989, pp. 381–384.Google Scholar