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Epsilon Entropy of Stochastic Processes

Edward C. Posner, Eugene R. Rodemich and Howard Rumsey, Jr.
The Annals of Mathematical Statistics
Vol. 38, No. 4 (Aug., 1967), pp. 1000-1020
Stable URL: http://www.jstor.org/stable/2238818
Page Count: 21
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Epsilon Entropy of Stochastic Processes
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Abstract

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 L2[ 0, 1] 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 L2[ 0, 1] 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 ∑ nσn 2 be finite, where σ1 2 ≥ σ2 2 ≥ ⋯ are the eigenvalues of the process.

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