On the Extreme Value Behavior of $\vartheta$-Expansions

TL;DR

This paper develops extreme value theory for $ heta$-expansions, establishing limit distributions for maxima in related stochastic processes, similar to classical continued fraction results.

Contribution

It introduces the first extreme value results for $ heta$-expansions, extending classical continued fraction theorems to this new setting.

Findings

Limit distribution of maxima in $ heta$-continued fraction processes established.

Analogous results to Galambos and Philipp for $ heta$-expansions proved.

Borel-Bernstein theorem is key in deriving these results.

Abstract

The main objective of this paper is to develop extreme value theory for $\vartheta$-expansions. We establish the limit distribution of the maximum value in a $\vartheta$-continued fraction mixing stationary stochastic process, along with some related results. These findings are analogous to the theorems of J. Galambos and W. Philipp for regular continued fractions. Additionally, we emphasize that a Borel-Bernstein type theorem plays a crucial role.