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

TL;DR

The paper develops extreme-value theory for $\vartheta$-expansions by leveraging the invariant $\gamma_{\vartheta}$ and the $\psi$-mixing structure of the digit process, and proves a Borel-Bernstein-type 0-1 law that governs large-digit occurrences. It then establishes a Fréchet limit for the maximal digit $L_N$ and derives iterated-logarithm results, providing sharp probabilistic descriptions of how large digits appear in $\vartheta$-expansions. Together, these results extend classical extreme-value theory from regular continued fractions to the generalized $\vartheta$-expansions under natural dynamical-system assumptions, offering precise asymptotic laws and growth rates with potential applications in random-number generation and dynamical systems.

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.

Theorems & Definitions (25)

• Lemma 2.1
• proof
• Lemma 2.2: Borel-Cantelli
• Theorem 3.1: Borel-Bernstein-type theorem
• proof
• Corollary 3.2
• proof
• Definition 4.1: $\psi$-mixing
• Lemma 4.2
• proof