Random Continued fractions: L\'evy constant and Chernoff-type estimate

TL;DR

This paper extends classical results on continued fractions to a stochastic setting, establishing a Le9vy-type theorem and Chernoff estimates for random continued fractions derived from ergodic and mixing processes.

Contribution

It introduces a Le9vy-type metric theorem and Chernoff-type estimates for random continued fractions with ergodic and mixing assumptions, generalizing classical deterministic results.

Findings

Established a Le9vy-type metric theorem for random continued fractions.

Derived Chernoff-type estimates under mixing and moment conditions.

Extended classical continued fraction results to stochastic processes.

Abstract

Given a stochastic process $\{A_n, n \geq 1\}$ taking values in natural numbers, the random continued fractions is defined as $[A_1, A_2, \cdots, A_n, \cdots]$ analogue to the continued fraction expansion of real numbers. Assume that $\{A_n, n \geq 1\}$ is ergodic and the expectation $E(\log A_1) < \infty$, we give a L\'evy-type metric theorem which covers that of real case presented by L\'evy in 1929. Moreover, a corresponding Chernoff-type estimate is obtained under the conditions $\{A_n, n \geq 1\}$ is $\psi$-mixing and for each $0< t< 1$, $E(A_1^t) < \infty$.