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.