ON THE LARGE DEVIATION FOR THE GCFε EXPANSION WHEN THE PARAMETER ε∈[−1, 1]
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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).
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).
참고문헌 (9)
- A. Dembo, "Large Deviations Techniquesand Applications" (1998), Springer-Verlag
- P. Erdos, "On Engel’s and Sylvester’s series", Ann. Univ. Sci. Budapest. Eotvos. Sect. Math, 1, pp.7~32 (1958)
- L. L. Fang, "Large and moderate deviations for modified Engel continued fractions", Statist. Probab. Lett, 98, pp.98~106 (2015)
- F. Schweiger, "Continued fraction with increasing digits", Oster Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, 212, pp.69~77 (2003)
- L. M. Shen, "Some metric properties on the GCF fraction expansion", J. Number Theory, 130(1), pp.1~9 (2010)
- S. R. S. Varadhan, "Large Deviations and Applications" (1984), SIAM
- T. Zhong, "Metrical properties for a class of continued fractions with increasing digits", J. Number Theory, 128(6), pp.1506~1515 (2008)
- T. Zhong, "Metrical property for GCF𝜖 expansion with the parameter function ∊(k) = c(k+1)", Int. J. Number Theory, 11(7), pp.2065~2072 (2015)
- L. J. Zhu, "On the large deviations for Engel’s, Sylvester’s series and Cantor’s products", Electron. Commun. Probab, 19(2), pp.1~9 (2014)