On the Extremal Theory of Continued Fractions

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)

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

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

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

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

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)

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)

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

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

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

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

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

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)

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

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

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

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)

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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