# Continued fractions, the Chen-Stein method and extreme value theory

## TL;DR

This paper applies probability, ergodic theory, and real analysis to improve bounds on the convergence rate of extreme values in continued fraction digit distributions, enhancing understanding of their asymptotic behavior.

## Contribution

It introduces new bounds for convergence rates in extreme value theory for continued fractions, utilizing the Chen-Stein method and ergodic theory techniques.

## Key findings

- Improved upper bounds for convergence rates in Doeblin-Iosifescu asymptotics.
- Enhanced understanding of the extremal behavior of continued fraction digits.
- Methodology applicable to order statistics and extremal point processes.

## 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.