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Infinite measure preserving transformations with compact first regeneration

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Abstract

We study ergodic infinite measure preserving transformations T possessing reference sets of finite measure for which the set of densities of the conditional distributions given a first return (or entrance) at time n is precompact in a suitable function space. Assuming regular variation of wandering rates, we establish versions of the Darling-Kac theorem and the arcsine laws for waiting times and for occupation times which apply to transformations with indifferent orbits and to random walks driven by Gibbs-Markov maps.

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References

  1. J. Aaronson, The asymptotic distributional behaviour of transformations preserving infinite measures, J. Analyse Math. 39 (1981), 203–234.

    MATH  MathSciNet  Google Scholar 

  2. J. Aaronson, Random f-expansions, Ann. Probab. 14 (1986), 1037–1057.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Aaronson, An Introduction to Infinite Ergodic Theory, Amer. Math. Soc., Providence, RI, 1997.

    MATH  Google Scholar 

  4. J. Aaronson and M. Denker, A local limit theorem for stationary processes in the domain of attraction of a normal distribution, in Asymptotic Methods in Probability and Statistics with Applications, Birkhä user, Boston, MA, 2001, pp. 215–223.

    Google Scholar 

  5. 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.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Aaronson, M. Denker and M. Urbanski, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), 495–548.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Aaronson, M. Denker, O. Sarig and R. Zweimuller, Aperiodicity of cocycles and conditional local limit theorems, Stoch. Dyn. 4 (2004), 31–62.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Aaronson, M. Thaler and R. Zweimüller, Occupation times of sets of infinite measure for ergodic transformations, Ergodic Theory Dynam. Systems 25 (2005), 959–976.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Benedicks and M. Misiurewicz, Absolutely continuous invariant measures for maps with flat tops, Publ. Math. IHES 69 (1989), 203–213.

    MATH  MathSciNet  Google Scholar 

  10. N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, 1989.

  11. D. A. Darling and M. Kac, On occupation times for Markoff processes, Trans. Amer. Math. Soc. 84 (1957), 444–458.

    Article  MATH  MathSciNet  Google Scholar 

  12. N. Dunford and J. T. Schwartz, Linear Operators I, Interscience, New York, 1958.

    MATH  Google Scholar 

  13. E. B. Dynkin, Some limit theorems for sums of independent random variables with infinite mathematical expectation, in Select. Transl. Math. Statist. and Probability 1, Inst.Math. Statist. and Amer. Math. Soc., Providence, RI, 1961, pp. 171–189.

    Google Scholar 

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

    MathSciNet  Google Scholar 

  15. P. Erdős and M. Kac, On the number of positive sums of independent random variables, Bull. Amer. Math. Soc. 53 (1947), 1011–1020.

    MathSciNet  Google Scholar 

  16. W. Feller, An Introduction to Probability Theory and its Applications, Vol. I, 3rd ed., Wiley, New York, 1968.

    MATH  Google Scholar 

  17. W. Feller, An introduction to Probability Theory and its Applications, Vol. II, Wiley, New York, 1966.

    MATH  Google Scholar 

  18. Y. Guivarc’h and J. Hardy, Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov, Ann. Inst. H. Poincaré Probab. Statist. 24 (1988), 73–98.

    MATH  MathSciNet  Google Scholar 

  19. G. Helmberg, Über konservative Transformationen, Math. Ann. 165 (1966), 44–61.

    Article  MATH  MathSciNet  Google Scholar 

  20. T. Kamae and M. Keane, A simple proof of the ratio ergodic theorem, Osaka J.Math. 34 (1997), 653–657.

    MATH  MathSciNet  Google Scholar 

  21. J. Korevaar, Tauberian theory. A Century of Developments, Springer-Verlag, Berlin, 2004.

    MATH  Google Scholar 

  22. U. Krengel, Ergodic Theorems, Walter de Gruyter & Co., Berlin, 1985.

    MATH  Google Scholar 

  23. J. Lamperti, An occupation time theorem for a class of stochastic processes, Trans. Amer.Math. Soc. 88 (1958), 380–387.

    Article  MATH  MathSciNet  Google Scholar 

  24. J. Lamperti, Some limit theorems for stochastic processes, J. Math. Mech. 7 (1958), 433–448.

    MATH  MathSciNet  Google Scholar 

  25. P. Lévy, Sur certains processus stochastiques homogenes, CompositioMath. 7 (1939), 283–339.

    MATH  Google Scholar 

  26. M. Lin, Mixing for Markov operators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 19 (1971), 231–243.

    Article  MathSciNet  Google Scholar 

  27. M. Nowak, On the finest Lebesgue topology on the space of essentially bounded measurable functions, Pacific J. Math. 140 (1989), 155–161.

    MATH  MathSciNet  Google Scholar 

  28. H. H. Schaefer, Topological Vector Spaces, Springer-Verlag, New York-Berlin, 1971.

    Google Scholar 

  29. F. Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer.Math. Soc. 82 (1956), 323–339.

    Article  MATH  MathSciNet  Google Scholar 

  30. F. Spitzer, Principles of Random Walk, 2nd ed., Springer-Verlag, New York-Heidelberg, 1976.

    MATH  Google Scholar 

  31. M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math. 37 (1980), 303–314.

    Article  MATH  MathSciNet  Google Scholar 

  32. M. Thaler, Transformations on [0, 1] with infinite invariant measures, Israel J.Math. 46 (1983), 67–96.

    Article  MATH  MathSciNet  Google Scholar 

  33. M. Thaler, A limit theorem for the Perron-Frobenius operator of transformations on [0, 1] with indifferent fixed points, Israel J. Math. 91 (1995), 111–127.

    Article  MATH  MathSciNet  Google Scholar 

  34. M. Thaler, The Dynkin-Lamperti arc-Sine laws for measure preserving transformations, Trans. Amer. Math. Soc. 350 (1998), 4593–4607.

    Article  MATH  MathSciNet  Google Scholar 

  35. M. Thaler, A limit theorem for sojourns near indifferent fixed points of one-dimensional maps, Ergodic Theory Dynam. Systems 22 (2002), 1289–1312.

    Article  MATH  MathSciNet  Google Scholar 

  36. M. Thaler and R. Zweimüller, Distributional limit theorems in infinite ergodic theory, Probab. Theory Relat. Fields 135 (2006), 15–52.

    Article  MATH  Google Scholar 

  37. D. V. Widder, The Laplace Transform, Princeton University Press, 1946.

  38. D. V. Widder, An Introduction to Transform Theory, Academic Press, New York, 1971.

    MATH  Google Scholar 

  39. P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Press, 1991.

  40. K. Yosida, Mean ergodic theorem in Banach spaces, Proc. Imp. Acad. Tokyo 14 (1938), 292–294.

    MATH  MathSciNet  Google Scholar 

  41. R. Zweimüller, Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points, Nonlinearity 11 (1998), 1263–1276.

    Article  MATH  MathSciNet  Google Scholar 

  42. R. Zweimüller, Ergodic properties of infinite measure preserving interval maps with indifferent fixed points, Ergodic Theory Dynam. Systems 20 (2000), 1519–1549.

    Article  MATH  MathSciNet  Google Scholar 

  43. R. Zweimüller, Hopf’s ratio ergodic theorem by inducing, Colloq.Math. 101 (2004), 289–292.

    MATH  MathSciNet  Google Scholar 

  44. R. Zweimüller, Kuzmin, coupling, cones, and exponential mixing, Forum Math. 16 (2004), 447–457.

    Article  MATH  MathSciNet  Google Scholar 

  45. R. Zweimüller, S-unimodal Misiurewicz maps with flat critical points, Fund.Math. 181 (2004), 1–25.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090, Wien, Austria

    Roland Zweimüller

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  1. Roland Zweimüller
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Correspondence to Roland Zweimüller.

Additional information

This research was supported by an APART [Austrian programme for advanced research and technology] fellowship of the Austrian Academy of Sciences. Much of this work was done at the Mathematics Department of Imperial College London. I also benefitted from a JRF at the ESI in Vienna.

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Zweimüller, R. Infinite measure preserving transformations with compact first regeneration. J Anal Math 103, 93–131 (2007). https://doi.org/10.1007/s11854-008-0003-y

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  • DOI: https://doi.org/10.1007/s11854-008-0003-y

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