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August, 1973 Epsilon Entropy of Stochastic Processes With Continuous Paths
Edward C. Posner, Eugene R. Rodemich
Ann. Probab. 1(4): 674-689 (August, 1973). DOI: 10.1214/aop/1176996894

Abstract

This paper shows that the epsilon entropy in the sup norm of a wide variety of processes with continuous paths on the unit interval is finite. In fact, the class coincides with the class of processes for which proofs of continuity have been given from a covariance condition. This suggests the conjecture that the epsilon entropy of any process continuous on the unit interval is finite in the sup norm of continuous functions. The epsilon entropy considered in this paper is defined as the minimum Shannon entropy of any partition by sets of diameter at most epsilon of the space of continuous functions on the unit interval, where the probability is the one inherited from the given process. The proof proceeds by constructing partitions and estimating their entropy using probability bounds.

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Edward C. Posner. Eugene R. Rodemich. "Epsilon Entropy of Stochastic Processes With Continuous Paths." Ann. Probab. 1 (4) 674 - 689, August, 1973. https://doi.org/10.1214/aop/1176996894

Information

Published: August, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0261.60045
MathSciNet: MR356207
Digital Object Identifier: 10.1214/aop/1176996894

Keywords: Automata , Circuits , continous processes , covariance conditions , Epsilon entropy , general theory (6040) , Information and Communication , Information Theory (9420) , modulus of continuity , partitions , Probability-stochastic processes , Wiener process

Rights: Copyright © 1973 Institute of Mathematical Statistics

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Vol.1 • No. 4 • August, 1973
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