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April, 1976 On the Minimum Number of Fixed Length Sequences with Fixed Total Probability
John C. Kieffer
Ann. Probab. 4(2): 335-337 (April, 1976). DOI: 10.1214/aop/1176996139

Abstract

Let $X_1, X_2,\cdots$ be a stationary sequence of $B$-valued random variables, where $B$ is a finite set. For each positive integer $n$, and number $\lambda$ such that $0 < \lambda < 1$, let $N(n, \lambda)$ be the cardinality of the smallest set $E \subset B^n$ such that $P\lbrack(X_1, X_2,\cdots, X_n) \in E\rbrack > 1 - \lambda$. An example is given to show that $\lim_{n\rightarrow\infty}n^{-1} \log N(n, \lambda)$ may not exist for some $\lambda$, thereby settling in the negative a conjecture of Parthasarathy.

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John C. Kieffer. "On the Minimum Number of Fixed Length Sequences with Fixed Total Probability." Ann. Probab. 4 (2) 335 - 337, April, 1976. https://doi.org/10.1214/aop/1176996139

Information

Published: April, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0337.60007
MathSciNet: MR406698
Digital Object Identifier: 10.1214/aop/1176996139

Subjects:
Primary: 60B05
Secondary: 28A35, 28A65, 94A15

Rights: Copyright © 1976 Institute of Mathematical Statistics

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Vol.4 • No. 2 • April, 1976
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