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A Note on Rényi's ‘Record’ Problem and Engel's Series

Part of: Markov processes Nonparametric inference Elementary number theory Limit theorems

Published online by Cambridge University Press:  15 February 2018

Lulu Fang  and
Min Wu
Lulu Fang*
Affiliation:
School of Mathematics, Sun Yat-sen University, Guangzhou 510275, P.R. China (fanglulu@mail.sysu.edu.cn)
Min Wu
Affiliation:
Department of Mathematics, South China University of Technology, Guangzhou 510640, P.R. China (wumin@scut.edu.cn)
*
*Corresponding author.
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Abstract

In 1973, Williams [D. Williams, On Rényi's ‘record’ problem and Engel's series, Bull. London Math. Soc.5 (1973), 235–237] introduced two interesting discrete Markov processes, namely C-processes and A-processes, which are related to record times in statistics and Engel's series in number theory respectively. Moreover, he showed that these two processes share the same classical limit theorems, such as the law of large numbers, central limit theorem and law of the iterated logarithm. In this paper, we consider the large deviations for these two Markov processes, which indicate that there is a difference between C-processes and A-processes in the context of large deviations.

Keywords

large deviationsMarkov processesrecord timesEngel's series

MSC classification

Primary: 60F10: Large deviations 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 62G30: Order statistics; empirical distribution functions 11A67: Other representations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 2 , May 2018 , pp. 363 - 369
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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