Volume 281, Issue 9 pp. 1294-1306

Original Paper

A distributional limit law for the continued fraction digit sum

Marc Kesseböhmer,

Corresponding Author

Marc Kesseböhmer

[email protected]

Universität Bremen, Fachbereich 3: Mathematik und Informatik, Bibliothekstrasse 1, 28359 Bremen, Germany

Phone: +49 421 218 8619, Fax: +49 421 218 4253Search for more papers by this author

Mehdi Slassi,

Mehdi Slassi

[email protected]

Universität Bremen, Fachbereich 3: Mathematik und Informatik, Bibliothekstrasse 1, 28359 Bremen, Germany

Phone: +49 421 218 9382, Fax: +49 421 218 4253

Search for more papers by this author

Marc Kesseböhmer,

Corresponding Author

Marc Kesseböhmer

[email protected]

Universität Bremen, Fachbereich 3: Mathematik und Informatik, Bibliothekstrasse 1, 28359 Bremen, Germany

Phone: +49 421 218 8619, Fax: +49 421 218 4253Search for more papers by this author

Mehdi Slassi,

Mehdi Slassi

[email protected]

Universität Bremen, Fachbereich 3: Mathematik und Informatik, Bibliothekstrasse 1, 28359 Bremen, Germany

Phone: +49 421 218 9382, Fax: +49 421 218 4253

Search for more papers by this author

First published: 13 August 2008

https://doi.org/10.1002/mana.200510679

Citations: 6

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Abstract

We consider the continued fraction digits as random variables measured with respect to Lebesgue measure. The logarithmically scaled and normalised fluctuation process of the digit sums converges strongly distributional to a random variable uniformly distributed on the unit interval. For this process normalised linearly we determine a large deviation asymptotic. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

References

[1] J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs Vol. 50 (American Mathematical Society, Providence, RI, 1997).
Google Scholar
[2] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications Vol. 27 (Cambridge University Press, Cambridge, 1989).
Google Scholar
[3] H. G. Diamond, and J. D. Vaaler, Estimates for partial sums of continued fraction partial quotients, Pacific J. Math. 122, No. 1, 73–82 (1986).
10.2140/pjm.1986.122.73
Web of Science® Google Scholar
[4] Y. Guivarc'h, and Y. Le Jan, Asymptotic winding of the geodesic flow on modular surfaces and continued fractions, Ann. Sci. École Norm. Sup. (4) 26, No. 1, 23–50 (1993).
Google Scholar
[5] Y. Guivarc'h, and Y. Le Jan, Note rectificative: “Asymptotic winding of the geodesic flow on modular surfaces and continued fractions” [Ann. Sci. École Norm. Sup. (4) 26, No. 1, 23–50; MR1209912 (94a:58157) (1993)], Ann. Sci. École Norm. Sup. (4) 29, No. 6, 811–814 (1996).
Google Scholar
[6] L. Heinrich, Rates of convergence in stable limit theorems for sums of exponentially ψ -mixing random variables with an application to metric theory of continued fractions, Math. Nachr. 131, 149–165 (1987).
10.1002/mana.19871310114
Web of Science® Google Scholar
[7] D. Hensley, The statistics of the continued fraction digit sum, Pacific J. Math. 192, No. 1, 103–120 (2000).
10.2140/pjm.2000.192.103
Web of Science® Google Scholar
[8] M. Kesseböhmer, and M. Slassi, Limit laws for distorted critical return time processes in infinite ergodic theory, Stoch. Dyn. 1, Nr. 7, 103–121 (2007).
10.1142/S0219493707001962
Web of Science® Google Scholar
[9] M. Kesseböhmer, and M. Slassi, Large deviation asymptotics for continued fraction expansions, Stoch. Dyn. 1, Nr. 8, 103–113 (2008).
10.1142/S0219493708002226
Web of Science® Google Scholar
[10] A. Ya. Khinchin, Continued Fractions (The University of Chicago Press, Chicago, Ill. – London, 1964).
Google Scholar
[11] P. Lévy, Fractions continues aléatoires, Rend. Circ. Mat. Palermo (2) 1, 170–208 (1952).
10.1007/BF02847786
Web of Science® Google Scholar
[12] W. Philipp, Limit theorems for sums of partial quotients of continued fractions, Monatsh. Math. 105, No. 3, 195–206 (1988).
10.1007/BF01636928
Web of Science® Google Scholar
[13] M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math. 37, No. 4, 303–314 (1980).
10.1007/BF02788928
Web of Science® Google Scholar
[14] M. Thaler, Transformations on [0, 1] with infinite invariant measures, Israel J. Math. 46, No. 1–2, 67–96 (1983).
10.1007/BF02760623
Web of Science® Google Scholar
[15] M. Thaler, A limit theorem for the Perron–Frobenius operator of transformations on [0, 1] with indifferent fixed points, Israel J. Math. 91, No. 1–3, 111–127 (1995).
10.1007/BF02761642
Web of Science® Google Scholar
[16] M. Thaler, The asymptotics of the Perron–Frobenius operator of a class of interval maps preserving infinite measures, Studia Math. 143, No. 2, 103–119 (2000).
Web of Science® Google Scholar

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Volume281, Issue9

September 2008

Pages 1294-1306

A distributional limit law for the continued fraction digit sum

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A distributional limit law for the continued fraction digit sum

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