Skip to main content

On the Extremal Theory of Continued Fractions

Abstract

Letting \(x=[a_1(x), a_2(x), \ldots ]\) denote the continued fraction expansion of an irrational number \(x\in (0, 1)\), Khinchin proved that \(S_n(x)=\sum \nolimits _{k=1}^n a_k(x) \sim \frac{1}{\log 2}n\log n\) in measure, but not for almost every \(x\). Diamond and Vaaler showed that, removing the largest term from \(S_n(x)\), the previous asymptotics will hold almost everywhere, this shows the crucial influence of the extreme terms of \(S_n (x)\) on the sum. In this paper we determine, for \(d_n\rightarrow \infty \) and \(d_n/n\rightarrow 0\), the precise asymptotics of the sum of the \(d_n\) largest terms of \(S_n(x)\) and show that the sum of the remaining terms has an asymptotically Gaussian distribution.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Arov, D.Z., Bobrov, A.A.: The extreme terms of a sample and their role in the sum of independent variables. Theor. Probab. Appl. 5, 377–396 (1960)

    Article  MathSciNet  Google Scholar 

  2. 2.

    Bickel, P.J., Wichura, M.J.: Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat. 42, 1656–1670 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  3. 3.

    Billingsley, P.: Ergodic Theory and Information. Wiley, New York (1965)

    MATH  Google Scholar 

  4. 4.

    Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)

    MATH  Google Scholar 

  5. 5.

    Bradley, R.: Introduction to Strong Mixing Conditions, vol. I. Kendrick Press, Heber City (2007)

    Google Scholar 

  6. 6.

    Csörgő, S., Horváth, L., Mason, D.: What portion of the sample makes a partial sum asymptotically stable or normal? Z. Wahrschein. verw. Gebiete 72, 1–16 (1986)

    Google Scholar 

  7. 7.

    Csörgő, S., Simons, G.: A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games. Stat. Probab. Lett. 26, 65–73 (1996)

    Article  Google Scholar 

  8. 8.

    Darling, D.: The influence of the maximum term in the addition of independent random variables. Trans. Am. Math. Soc. 73, 95–107 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  9. 9.

    Diamond, H., Vaaler, J.: Estimates for partial sums of continued fraction partial quotients. Pac. J. Math. 122, 73–82 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. 10.

    Doeblin, W.: Remarques sur la théorie métrique des fractions continues. Compos. Math. 7, 353–371 (1940)

    MathSciNet  Google Scholar 

  11. 11.

    Galambos, J.: The distribution of the largest coefficient in continued fraction expansions. Q. J. Math. Oxf. Ser. 23, 147–151 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  12. 12.

    Galambos, J.: An iterated logarithm type theorem for the largest coefficient in continued fractions. Acta Arith. 25, 359–364 (1973/74)

  13. 13.

    Gordin, M.I., Reznik, M.H.: The law of the iterated logarithm for the denominators of continued fractions. Vestn. Leningr. Univ. 25, 28–33 (1970). (In Russian)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Heinrich, L.: Rates of convergence in stable limit theorems for sums of exponentially \(\psi \)- mixing random variables with an application to metric theory of continued fractions. Math. Nachr. 131, 149–165 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. 15.

    Ibragimov, I.A.: A theorem from the metric theory of continued fractions. Vestn. Leningr. Univ. 1, 13–24 (1960). (In Russian)

    Google Scholar 

  16. 16.

    Iosifescu, M.: A Poisson law for \(\psi \)-mixing sequences establishing the truth of a Doeblin’s statement. Rev. Roum. Math. Pures Appl. 22, 1441–1447 (1977)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Iosifescu, M.: A survey of the metric theory of continued fractions, fifty years after Doeblin’s 1940 paper. In: Grigelionis, B., et al. (eds.) Probability Theory and Mathematical Statistics, vol. 1, pp. 550–572. Mokslas, Vilnius (1990)

    Google Scholar 

  18. 18.

    Khinchin, A.J.: Metrische Kettenbruchprobleme. Compos. Math. 1, 361–382 (1935)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Kusmin, R.: Sur un problème de Gauss. Atti Congr. Int. Bol. 6, 83–89 (1928)

    Google Scholar 

  20. 20.

    Leadbetter, M., Rootzén, H.: Extremal theory for stochastic processes. Ann. Probab. 16, 431–478 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  21. 21.

    Lévy, P.: Sur les lois de probabilité dont dépendent les quotients complets et incomplets d’une fraction continue. Bull. Sci. Math. Fr. 57, 178–194 (1929)

    MATH  Google Scholar 

  22. 22.

    Lévy, P.: Fractions continues aléatoires. Rend. Circ. Mat. Palermo 1, 170–208 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  23. 23.

    Philipp, W.: Some metrical theorems in number theory II. Duke Math. J. 37, 447–458 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  24. 24.

    Philipp, W.: A conjecture of Erdős on continued fractions. Acta Arith. 28, 379–386 (1975/76)

  25. 25.

    Philipp, W.: Limit theorems for partial quotients of continued fractions. Mon. Math. 105, 195–206 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  26. 26.

    Philipp, W., Stackelberg, O.: Zwei Grenzwertssätze für Kettenbrüche. Math. Ann. 181, 152–156 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  27. 27.

    Samur, J.: On some limit theorems for continued fractions. Trans. Am. Math. Soc. 316, 53–79 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  28. 28.

    Samur, J.: Some remarks on a probability limit theorem for continued fractions. Trans. Am. Math. Soc. 348, 1411–1428 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. 29.

    Stackelberg, O.P.: On the law of the iterated logarithm for continued fractions. Duke Math. J. 33, 801–819 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  30. 30.

    Szewczak, Z.S.: On limit theorems for continued fractions. J. Theor. Probab. 22, 239–255 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. 31.

    Utev, S.A.: On the central limit theorem for \(\varphi \)-mixing arrays of random variables. Theory Probab. Appl. 35, 131–139 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  32. 32.

    Vardi, I.: The St. Petersburg game and continued fractions. C. R. Acad. Sci. Paris Ser. I Math. 324, 913–918 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We would like to thank the referee for her/his remarks leading to a substantial improvement of the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to István Berkes.

Additional information

Alina Bazarova received research support from Austrian Science Fund (FWF) Grant W1230.

István Berkes received research support from Austrian Science Fund (FWF) Grant P24302-N18 and from OTKA Grant K 108615.

Lajos Horváth received research support from NSF Grant DMS-13-05858.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bazarova, A., Berkes, I. & Horváth, L. On the Extremal Theory of Continued Fractions. J Theor Probab 29, 248–266 (2016). https://doi.org/10.1007/s10959-014-0577-5

Download citation

Keywords

  • Continued fraction expansion
  • Extreme elements
  • Mixing random variables
  • Central limit theorem

Mathematics Subject Classification (2010)

  • Primary 11K50
  • 60F05
  • 60G70