## The Annals of Mathematical Statistics

### Product Entropy of Gaussian Distributions

#### 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 $k$th eigenfunction axis, a partition by intervals of length $\epsilon_k$ 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 $\mathbf{\sum} \mathbf{\epsilon}_k^2 \leqq \mathbf{\epsilon}^2$ 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 $\mathbf{\sum} \lambda_k \log \lambda_k^{-1}$ is finite, where $\lambda_k$ is the $k$th eigenvalue of the process. When the above series is finite, the values of $\mathbf{\epsilon}_k$ 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

#### Article information

Source
Ann. Math. Statist., Volume 40, Number 3 (1969), 870-904.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177697595

Digital Object Identifier
doi:10.1214/aoms/1177697595

Mathematical Reviews number (MathSciNet)
MR248954

Zentralblatt MATH identifier
0308.60020

JSTOR