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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Dynkin-Lamperti arc-sine laws
for measure preserving transformations


Author: Maximilian Thaler
Journal: Trans. Amer. Math. Soc. 350 (1998), 4593-4607
MSC (1991): Primary 28D05, 60F05, 60K05
DOI: https://doi.org/10.1090/S0002-9947-98-02312-5
MathSciNet review: 1603998
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Abstract: Arc-sine laws in the sense of renewal theory are proved for return time processes generated by transformations with infinite invariant measure on sets satisfying a type of Darling-Kac condition, and an application to real transformations with indifferent fixed points is discussed.


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  • 1. J. Aaronson, The asymptotic distributional behaviour of transformations preserving infinite measures, J. Analyse Math. 39 (1981), 203-234 MR 82m:28030
  • 2. J. Aaronson, Random f-expansions, Ann. Probab. 14 (1986), 1037-1057 MR 87k:60057
  • 3. J. Aaronson, An introduction to infinite ergodic theory, AMS, 1997 CMP 97:13
  • 4. N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular variation, Cambridge University Press, Cambridge 1987 MR 88i:26004
  • 5. D.A. Darling and M. Kac, On occupation times for Markoff processes, Trans. Amer. Math. Soc. 84 (1957), 444-458 MR 18:832a
  • 6. E.B. Dynkin, Some limit theorems for sums of independent random variables with infinite mathematical expectations, Selected Transl. in Math. Statist. and Probability 1 (1961), 171-189 MR 22:7164
  • 7. W. Feller, An introduction to probability theory and its applications, Vol. II. John Wiley & Sons, New York 1971 MR 42:5292
  • 8. J. Lamperti, An occupation time theorem for a class of stochastic processes, Trans. Amer. Math. Soc. 88 (1958), 380-387 MR 20:1372
  • 9. J. Lamperti, Some limit theorems for stochastic processes, J. Math. Mech. 7 (1958), 433-448 MR 20:4888
  • 10. P. Manneville, Intermittency, self-similarity and $1/f$ spectrum in dissipative dynamical systems, J. Physique 41 (1980), 1235-1243 MR 82e:58065
  • 11. F. Schweiger, Ergodic theory of fibered systems and metric number theory, Clarendon Press, Oxford 1995 MR 97h:11083
  • 12. R.S. Slack, Further notes on branching processes with mean one, Z. Wahrscheinlichkeitstheorie verw. Geb. 25 (1972), 31-38 MR 48:9871
  • 13. M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Isr. J. Math. 37 (1980), 303-314 MR 82f:28021
  • 14. M. Thaler, Transformations on $[0,1]$ with infinite invariant measures, Isr. J. Math. 46 (1983), 67-96 MR 85g:28020
  • 15. M. Thaler and C. Reichsöllner, Arc sine type limit laws for interval mappings, Manuscript, Salzburg 1986
  • 16. M. Thaler, A limit theorem for the Perron-Frobenius operator of transformations on $[0,1]$ with indifferent fixed points, Isr. J. Math. 91 (1995), 111-127 MR 96i:28020
  • 17. M. Thaler, The invariant densities for maps modeling intermittency, J. Statist. Phys. 79 (1995), 739-741 MR 96a:58119
  • 18. R. Zweimüller, Probabilistic properties of dynamical systems with infinite invariant measure, Diplomarbeit, Salzburg 1995

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Additional Information

Maximilian Thaler
Affiliation: Institute of Mathematics University of Salzburg Hellbrunnerstraße 34 A-5020 Salzburg, Austria
Email: Maximilian.Thaler@sbg.ac.at

DOI: https://doi.org/10.1090/S0002-9947-98-02312-5
Received by editor(s): October 29, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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