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Continued fractions, the Chen–Stein method and extreme value theory

  • ANISH GHOSH (a1), MAXIM SØLUND KIRSEBOM (a2) and PARTHANIL ROY (a3)

Abstract

In this work we deal with extreme value theory in the context of continued fractions using techniques from probability theory, ergodic theory and real analysis. We give an upper bound for the rate of convergence in the Doeblin–Iosifescu asymptotics for the exceedances of digits obtained from the regular continued fraction expansion of a number chosen randomly from $(0,1)$ according to the Gauss measure. As a consequence, we significantly improve the best known upper bound on the rate of convergence of the maxima in this case. We observe that the asymptotics of order statistics and the extremal point process can also be investigated using our methods.

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[1] Aaronson, J.. An Introduction to Infinite Ergodic Theory. American Mathematical Society, Providence, RI, 1997.
[2] Arratia, R., Goldstein, L. and Gordon, L.. Two moments suffice for Poisson approximations: the Chen–Stein method. Ann. Probab. 17 (1989), 925.
[3] Chang, Y. and Ma, J.. Some distribution results of the Oppenheim continued fractions. Monatsh. Math. 184(3) (2017), 379399.
[4] Chiarini, A., Cipriani, A. and Hazra, R. S.. A note on the extremal process of the supercritical Gaussian free field. Electron. Commun. Probab. 20 (2015), paper no. 74.
[5] Davis, R.. Stable limits for partial sums of dependent random variables. Ann. Probab. 11(2) (1983), 262269.
[6] Davis, R. and Hsing, T.. Point processes for partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23(2) (1995), 879917.
[7] Doeblin, W.. Remarques sur la théorie métrique des fractions continues. Compos. Math. 7 (1940), 353371.
[8] Freedman, D.. The Poisson approximation for dependent events. Ann. Probab. 2 (1974), 256269.
[9] Galambos, J.. The distribution of the largest coefficient in continued fraction expansions. Quart. J. Math. 23(2) (1972), 147151.
[10] Iosifescu, M.. A Poisson law for 𝜓-mixing sequences establishing the truth of a Doeblin’s statement. Rev. Roumaine Math. Pures Appl. 22 (1977), 14411447.
[11] Kallenberg, O.. Random Measures, 3rd edn. Akademie-Verlag, Berlin, 1983.
[12] Khintchine, A.. Continued Fractions. University of Chicago Press, Chicago, 1964.
[13] Nakada, H. and Natsui, R.. On the metrical theory of continued fraction mixing fibred systems and its application to Jacobi–Perron algorithm. Monatsh. Math. 138 (2003), 267288.
[14] Philipp, W.. A conjecture of Erdös on continued fractions. Acta Arith. 28(4) (1976), 379386.
[15] Resnick, S.. Extreme Values, Regular Variation and Point Processes. Springer, New York, 1987.
[16] Resnick, S.. Heavy-Tail Phenomena. Springer, New York, 2007.
[17] Resnick, S. and de Haan, L.. Second-order regular variation and rates of convergence in extreme-value theory. Ann. Probab. 24(1) (1989), 97124.
[18] Smith, R. L.. Extreme value theory for dependent sequences via the Stein–Chen method of Poisson approximation. Stochastic Process. Appl. 30(2) (1988), 317327.
[19] Tyran-Kamińska, M.. Weak convergence to Lévy stable processes in dynamical systems. Stoch. Dyn. 10(2) (2010), 263289.

Keywords

MSC classification

Continued fractions, the Chen–Stein method and extreme value theory

  • ANISH GHOSH (a1), MAXIM SØLUND KIRSEBOM (a2) and PARTHANIL ROY (a3)

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