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

Thaler, Maximilian

doi:10.1090/S0002-9947-98-02312-5

The Dynkin-Lamperti arc-sine laws for measure preserving transformations
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Trans. Amer. Math. Soc. 350 (1998), 4593-4607 Request permission

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.

References

Jon Aaronson, The asymptotic distributional behaviour of transformations preserving infinite measures, J. Analyse Math. 39 (1981), 203–234. MR 632462, DOI 10.1007/BF02803336
Jon. Aaronson, Random $f$-expansions, Ann. Probab. 14 (1986), no. 3, 1037–1057. MR 841603
J. Aaronson, An introduction to infinite ergodic theory, AMS, 1997
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871, DOI 10.1017/CBO9780511721434
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
E. B. Dynkin, Some limit theorems for sums of independent random variables with infinite mathematical expectations, Select. Transl. Math. Statist. and Probability, Vol. 1, Inst. Math. Statist. and Amer. Math. Soc., Providence, R.I., 1961, pp. 171–189. MR 0116376
William Feller, An introduction to probability theory and its applications. Vol. II. , 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
John Lamperti, An occupation time theorem for a class of stochastic processes, Trans. Amer. Math. Soc. 88 (1958), 380–387. MR 94863, DOI 10.1090/S0002-9947-1958-0094863-X
John Lamperti, Some limit theorems for stochastic processes, J. Math. Mech. 7 (1958), 433–448. MR 0098429, DOI 10.1512/iumj.1958.7.57027
P. Manneville, Intermittency, self-similarity and $1/f$ spectrum in dissipative dynamical systems, J. Physique 41 (1980), no. 11, 1235–1243 (English, with French summary). MR 592697, DOI 10.1051/jphys:0198000410110123500
Fritz Schweiger, Ergodic theory of fibred systems and metric number theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. MR 1419320
R. S. Slack, Further notes on branching processes with mean $1$, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972/73), 31–38. MR 331539, DOI 10.1007/BF00533333
Maximilian Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math. 37 (1980), no. 4, 303–314. MR 599464, DOI 10.1007/BF02788928
Maximilian Thaler, Transformations on $[0,\,1]$ with infinite invariant measures, Israel J. Math. 46 (1983), no. 1-2, 67–96. MR 727023, DOI 10.1007/BF02760623
M. Thaler and C. Reichsöllner, Arc sine type limit laws for interval mappings, Manuscript, Salzburg 1986
Maximilian Thaler, A limit theorem for the Perron-Frobenius operator of transformations on $[0,1]$ with indifferent fixed points, Israel J. Math. 91 (1995), no. 1-3, 111–127. MR 1348308, DOI 10.1007/BF02761642
Maximilian Thaler, The invariant densities for maps modeling intermittency, J. Statist. Phys. 79 (1995), no. 3-4, 739–741. MR 1327905, DOI 10.1007/BF02184879
R. Zweimüller, Probabilistic properties of dynamical systems with infinite invariant measure, Diplomarbeit, Salzburg 1995