Product Entropy of Gaussian Distributions

This paper studies the product epsilon entropy of mean-continuous Gaussian processes. That is, a given mean-continuous Gaussian process on the unit interval is expanded into its Karhunen expansion. Along the kth eigenfunction axis, a partition by intervals of length εkis made, and the entropy of the...

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Bibliographic Details
Published in:The Annals of mathematical statistics 1969-06, Vol.40 (3), p.870-904
Main Authors: Posner, Edward C., Rodemich, Eugene R., Rumsey, Howard
Format: Article
Language:English
Subjects:
Eigenvalues
Entropy
Mathematical functions
Mathematical intervals
Mathematics
Online Access:Get full text
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Summary:This paper studies the product epsilon entropy of mean-continuous Gaussian processes. That is, a given mean-continuous Gaussian process on the unit interval is expanded into its Karhunen expansion. Along the kth eigenfunction axis, a partition by intervals of length εkis made, and the entropy of the resulting discrete distribution is noted. The infimum of the sum over k of these entropies subject to the constraint that ∑ εk 2≤ ε2is the product epsilon entropy of the process. It is shown that the best partition to take along each eigenfunction axis is the one in which 0 is the midpoint of an interval in the partition. Furthermore, the product epsilon entropy is finite if and only if ∑ λklog λk -1is finite, where λkis the kth eigenvalue of the process. When the above series is finite, the values of εkwhich achieve the product entropy are found. Asymptotic expressions for the product epsilon entropy are derived in some special cases. The problem arises in the theory of data compression, which studies the efficient representation of random data with prescribed accuracy
ISSN:0003-4851
2168-8990
DOI:10.1214/aoms/1177697595