The Annals of Probability

On the Minimum Number of Fixed Length Sequences with Fixed Total Probability

John C. Kieffer

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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.

Article information

Source
Ann. Probab. Volume 4, Number 2 (1976), 335-337.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996139

Digital Object Identifier
doi:10.1214/aop/1176996139

Mathematical Reviews number (MathSciNet)
MR406698

Zentralblatt MATH identifier
0337.60007

JSTOR
links.jstor.org

Subjects
Primary: 60B05: Probability measures on topological spaces
Secondary: 28A65 28A35: Measures and integrals in product spaces 94A15: Information theory, general [See also 62B10, 81P94]

Keywords
Stationary measures Shannon-McMillan theorem shift transformation on a product space

Citation

Kieffer, John C. On the Minimum Number of Fixed Length Sequences with Fixed Total Probability. Ann. Probab. 4 (1976), no. 2, 335--337. doi:10.1214/aop/1176996139. https://projecteuclid.org/euclid.aop/1176996139


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