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

 
 

 

Some remarks on a probability limit theorem
for continued fractions


Author: Jorge D. Samur
Journal: Trans. Amer. Math. Soc. 348 (1996), 1411-1428
MSC (1991): Primary 11K50, 60F17; Secondary 11K60, 60F05, 60F15
DOI: https://doi.org/10.1090/S0002-9947-96-01571-1
MathSciNet review: 1344212
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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that if a certain condition on the variances of the partial sums is satisfied then a theorem of Philipp and Stout, which implies the asymptotic fluctuation results known for independent random variables, can be applied to some quantities related to continued fractions. Previous results on the behavior of the approximation by the continued fraction convergents to a random real number are improved.


References [Enhancements On Off] (What's this?)

  • 1. P. Billingsley, Ergodic Theory and Information, Wiley, New York, 1965. MR 33:254
  • 2. P. Billingsley, Convergence of probability measures, Wiley, New York, 1968. MR 38:1718
  • 3. W. Bosma, H. Jager and F. Wiedijk, Some metrical observations on the approximation by continued fractions, Indag. Math. 45 (1983), 281--299. MR 85f:11059
  • 4. K. L. Chung, A course in Probability Theory, 2nd edition, Academic Press, New York, 1974. MR 49:11579
  • 5. W. Doeblin, Remarques sur la théorie métrique des fractions continues, Compositio Math. 7 (1940), 353--371. MR 2:107e
  • 6. M. I. Gordin, On the behavior of the variances of sums of random variables forming a stationary process, Theory Probab. Appl. 16 (1971), 474--484. MR 44:4809
  • 7. M. I. Gordin and M. H. Reznik, The law of the iterated logarithm for the denominators of continued fractions, Vestnik Leningrad Univ. 25 (1970), 28--33. (Russian) MR 43:1939
  • 8. S. Grigorescu and M. Iosifescu, Dependence with Complete Connections and its Applications, Cambridge University Press, Cambridge, 1990. MR 91j:60098
  • 9. I. A. Ibragimov, A metrical theorem in the theory of continued fractions, Vestnik Leningrad Univ. 1 (1961), 13--24. (Russian) MR 24:A3445
  • 10. I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen, 1971. MR 48:1287
  • 11. M. Iosifescu, On mixing coefficients for the continued fraction expansion, Stud. Cerc. Mat. 41 (1989), 491--499. MR 91h:11079
  • 12. D. E. Knuth, The Art of Computer Programming, Vol. 2. Seminumerical Algorithms, Addison-Wesley, Reading, Mass., 1981. MR 83i:68003
  • 13. D. E. Knuth, The distribution of continued fraction approximations, J. Number Theory 19 (1984), 443--448. MR 86d:11058
  • 14. P. Lévy, Théorie de l'addition des variables aléatoires, Gauthier-Villars, Paris, 1954.
  • 15. G. Misevi\v{c}ius, The evaluation of the remainder term in the limit theorem for functions of the elements of continued fractions, Lietuvos Matematikos Rinkinys 10 (1970), 293--308. (Russian) MR 45:5102
  • 16. W. Philipp and O. P. Stackelberg, Zwei Grenzwertsätze für Kettenbrüche, Math. Ann. 181 (1969), 152--156. MR 39:5503
  • 17. W. Philipp and W. Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc. 161 (1975), Errata (preprint, 1975). MR 55:6570
  • 18. J. D. Samur, On some limit theorems for continued fractions, Trans. Amer. Math. Soc. 316 (1989), 53--79. MR 90b:60030
  • 19. J. D. Samur, A functional central limit theorem in Diophantine approximation, Proc. Amer. Math. Soc. 111 (1991), 901--911. MR 91g:11087

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Additional Information

Jorge D. Samur
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Casilla de correo 172, 1900 La Plata, Argentina
Email: jorge@mate.unlp.edu.ar

DOI: https://doi.org/10.1090/S0002-9947-96-01571-1
Keywords: Continued fractions, approximation by the principal convergents, almost sure invariance principle
Received by editor(s): January 1, 1995
Additional Notes: Some results were announced at the 8th International Conference on Probability in Banach Spaces, Brunswick, Maine, July 1991.
Article copyright: © Copyright 1996 American Mathematical Society

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