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Product Entropy of Gaussian Distributions

Edward C. Posner, Eugene R. Rodemich and Howard Rumsey, Jr.
The Annals of Mathematical Statistics
Vol. 40, No. 3 (Jun., 1969), pp. 870-904
Stable URL: http://www.jstor.org/stable/2239636
Page Count: 35
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Product Entropy of Gaussian Distributions
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Abstract

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 εk is 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 ≤ ε2 is 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 ∑ λk log λk -1 is finite, where λk is the kth eigenvalue of the process. When the above series is finite, the values of εk which 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

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