According to Wikipedia, a continued fraction is “an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.” It has connections to various areas of mathematics including algebra, analysis, combinatorics, dynamical systems, number theory, probability theory, etc.
Continued fractions of numbers in the open interval (0,1) naturally gives rise to a dynamical system that dates back to Gauss and raises many interesting questions. We will concentrate on the extreme value theoretic aspects of this dynamical system. The first work in this direction was carried out by the famous French-German mathematician Doeblin (1940), who rightly observed that exceedances of this dynamical system have Poissonian asymptotics. However, his proof (carried out at the backdrop of World War II) had a subtle error, which was corrected more than three decades later by Iosifescu (1977). Meanwhile, Galambos (1972) had established that the scaled maxima of this dynamical system converges to the Frechet distribution.
After a detailed review of these results and the melancholic history of Wolfgang Doeblin, we will use a powerful Chen-Stein method (of establishing Poisson approximation for dependent Bernoulli random variables due to Arratia, Goldstein and Gordon (1989)) to give upper bounds on the rate of convergence in the Doeblin-Iosifescu asymptotics. We will also discuss consequences of our result and its connections to other important dynamical systems.
This talk is based on a joint work with Anish Ghosh (TIFR Mumbai) and Maxim Sølund Kirsebom (Univ of Hamburg).