Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2017; 54(3): 835-845

Online first article December 27, 2016      Printed May 1, 2017

https://doi.org/10.4134/JKMS.j160279

Copyright © The Korean Mathematical Society.

On the large deviation for the GCF$_\epsilon$ expansion when the parameter $\epsilon \in[-1,1]$

Ting Zhong

Jishou University

Abstract

The GCF$_\epsilon$ expansion is a new class of continued fractions induced by the transformation $T_{\epsilon}:(0,1]\to (0,1]$: $$ T_{\epsilon}(x)=\frac{-1+(k+1)x}{1+k-k\epsilon x} \ {\text{for}}\ x\in \big(1/(k+1), 1/k\big]. $$ \noindent Under the algorithm $T_{\epsilon}$, every $x\in (0,1]$ corresponds to an increasing digits sequences $\{k_n,{n\ge1}\}$. Their basic properties, including the ergodic properties, law of large number and central limit theorem have been discussed in \cite{Sc}, \cite{SZ} and \cite{Zh}. In this paper, we study the large deviation for the GCF$_\epsilon$ expansion and show that: $\big\{\frac{1}{n}\log k_n, n\ge1\big\}$ satisfies the different large deviation principles when the parameter $\epsilon$ changes in $[-1,1]$, which generalizes a result of L.~J.~Zhu \cite{Zhu} who considered a case when $\epsilon(k) \equiv 0$ (i.e., Engel series).

Keywords: large deviation principle, GCF$_\epsilon$ algorithm, parameter function $\epsilon(k)$

MSC numbers: 60F10, 11A67, 11K55

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