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ON THE LARGE DEVIATION FOR THE GCF𝝐 EXPANSION WHEN THE PARAMETER 𝝐 ∈ [-1, 1]

  • Zhong, Ting
  • Received : 2016.04.22
  • Published : 2017.05.01

Abstract

The $GCF_{\epsilon}$ expansion is a new class of continued fractions induced by the transformation $T_{\epsilon}:(0, 1]{\rightarrow}(0, 1]$: $T_{\epsilon}(x)={\frac{-1+(k+1)x}{1+k-k{\epsilon}x}}$ for $x{\in}(1/(k+1),1/k]$. Under the algorithm $T_{\epsilon}$, every $x{\in}(0,1]$ corresponds to an increasing digits sequences $\{k_n,n{\geq}1\}$. Their basic properties, including the ergodic properties, law of large number and central limit theorem have been discussed in [4], [5] and [7]. In this paper, we study the large deviation for the $GCF_{\epsilon}$ expansion and show that: $\{{\frac{1}{n}}{\log}k_n,n{\geq}1\}$ satisfies the different large deviation principles when the parameter ${\epsilon}$ changes in [-1, 1], which generalizes a result of L. J. Zhu [9] who considered a case when ${\epsilon}(k){\equiv}0$ (i.e., Engel series).

Keywords

large deviation principle, $GCF_{\epsilon}$ algorithm, parameter function ${\epsilon}(k)$

References

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Acknowledgement

Supported by : National Natural Science Foundation of China