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August, 1967 Epsilon Entropy of Stochastic Processes
Edward C. Posner, Eugene R. Rodemich, Howard Rumsey Jr.
Ann. Math. Statist. 38(4): 1000-1020 (August, 1967). DOI: 10.1214/aoms/1177698768


This paper introduces the concept of epsilon-delta entropy for "probabilistic metric spaces." The concept arises in the study of efficient data transmission, in other words, in "Data Compression." In a case of particular interest, the space is the space of paths of a stochastic process, for example $L_2\lbrack 0, 1\rbrack$ under the probability distribution induced by a mean-continuous process on the unit interval. For any epsilon and delta both greater than zero, the epsilon-delta entropy of any probabilistic metric space is finite. However, when delta is zero, the resulting entropy, called simply the epsilon entropy of the space, can be infinite. We give a simple condition on the eigenvalues of a process on $L_2\lbrack 0, 1\rbrack$ such that any process satisfying that condition has finite epsilon entropy for any epsilon greater than zero. And, for any set of eigenvalues not satisfying the given condition, we produce a mean-continuous process on the unit interval having infinite epsilon entropy for every epsilon greater than zero. The condition is merely that $\sum n\sigma_n^2$ be finite, where $\sigma_1^2 \geqq \sigma_2^2 \geqq \cdots$ are the eigenvalues of the process.


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Edward C. Posner. Eugene R. Rodemich. Howard Rumsey Jr.. "Epsilon Entropy of Stochastic Processes." Ann. Math. Statist. 38 (4) 1000 - 1020, August, 1967.


Published: August, 1967
First available in Project Euclid: 27 April 2007

zbMATH: 0168.18003
MathSciNet: MR211457
Digital Object Identifier: 10.1214/aoms/1177698768

Rights: Copyright © 1967 Institute of Mathematical Statistics

Vol.38 • No. 4 • August, 1967
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