Some remarks on a probability limit theorem for continued fractions
HTML articles powered by AMS MathViewer
- by Jorge D. Samur PDF
- Trans. Amer. Math. Soc. 348 (1996), 1411-1428 Request permission
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
- Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0192027
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
- W. Bosma, H. Jager, and F. Wiedijk, Some metrical observations on the approximation by continued fractions, Nederl. Akad. Wetensch. Indag. Math. 45 (1983), no. 3, 281–299. MR 718069
- Kai Lai Chung, A course in probability theory, 2nd ed., Probability and Mathematical Statistics, Vol. 21, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0346858
- Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
- M. I. Gordin, The behavior of the dispersion of sums of random variables that generate a stationary process, Teor. Verojatnost. i Primenen. 16 (1971), 484–494 (Russian, with English summary). MR 0287606
- M. I. Gordin and M. H. Reznik, The law of the iterated logarithm for the denominators of continued fractions, Vestnik Leningrad. Univ. 25 (1970), no. 13, 28–33 (Russian, with English summary). MR 0276191
- Marius Iosifescu and Şerban Grigorescu, Dependence with complete connections and its applications, Cambridge Tracts in Mathematics, vol. 96, Cambridge University Press, Cambridge, 1990. MR 1070097
- I. A. Ibragimov, A theorem from the metric theory of continued fractions, Vestnik Leningrad. Univ. 16 (1961), no. 1, 13–24 (Russian, with English summary). MR 0133619
- I. A. Ibragimov and Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov; Translation from the Russian edited by J. F. C. Kingman. MR 0322926
- Marius Iosifescu, On mixing coefficients for the continued fraction expansion, Stud. Cerc. Mat. 41 (1989), no. 6, 491–499. MR 1043848
- Donald E. Knuth, The art of computer programming. Vol. 2, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms. MR 633878
- Donald E. Knuth, The distribution of continued fraction approximations, J. Number Theory 19 (1984), no. 3, 443–448. MR 769794, DOI 10.1016/0022-314X(84)90083-0
- P. Lévy, Théorie de l’addition des variables aléatoires, Gauthier-Villars, Paris, 1954.
- G. A. Misjavičus, Estimation of the remainder terms in limit theorems for a distribution of functions of the elements of continued fractions, Litovsk. Mat. Sb. 10 (1970), 293–308 (Russian, with Lithuanian and English summaries). MR 0296041
- Walter Philipp and Olaf P. Stackelberg, Zwei Grenzwertsätze für Kettenbrüche, Math. Ann. 181 (1969), 152–156 (German). MR 244186, DOI 10.1007/BF01350634
- Walter Philipp and William Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc. 2 (1975), no. 161,, 161, iv+140. MR 433597, DOI 10.1090/memo/0161
- Jorge D. Samur, On some limit theorems for continued fractions, Trans. Amer. Math. Soc. 316 (1989), no. 1, 53–79. MR 948197, DOI 10.1090/S0002-9947-1989-0948197-9
- Jorge D. Samur, A functional central limit theorem in Diophantine approximation, Proc. Amer. Math. Soc. 111 (1991), no. 4, 901–911. MR 998739, DOI 10.1090/S0002-9939-1991-0998739-7
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
- 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.
- © Copyright 1996 American Mathematical Society
- 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