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 analysis, number theory, dynamical systems and probability theory.
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 had a subtle error, which was corrected much 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, we will discuss a powerful Stein-Chen method (due to Arratia, Goldstein and Gordon (1989)) of establishing Poisson approximation for dependent Bernoulli random variables. The final part of the talk will be about the application of this Stein-Chen technique 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 portion of the talk will be based on a joint work with Maxim Sølund Kirsebom and Anish Ghosh.
Familiarity with standard (discrete and absolutely continuous) probability distributions will be sufficient for this talk.