Open Access
April, 1990 Universal Almost Sure Data Compression
Donald S. Ornstein, Paul C. Shields
Ann. Probab. 18(2): 441-452 (April, 1990). DOI: 10.1214/aop/1176990840


An $n$-code is a mapping $c_n$ from the set $A^n$ of sequences of length $n$ drawn from a finite set $A$ into the set of finite length binary sequences $B^\ast$. A decoder with distortion $D$ is a map from $B^\ast$ back into $A^n$ that sends $c_n(a^n_1)$ into a sequence that agrees with $a^n_1$ in all but at most $Dn$ places. We describe a sequence of codes and associated decoders of distortion $D$ such that, for almost every sequence from an ergodic process, the number of bits per $A$-symbol converges almost surely to $R(D)$, the optimal compression attainable for the process. The codes are universal in that the statistics of the process need not be known in advance. Expected value results of this type were first obtained by Davisson and, independently, Fittinghof; almost sure results for the invertible case $(D = 0)$ are implicitly contained in the Ziv-Lempel algorithm. Our results also apply, virtually without change in proof, to random fields.


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Donald S. Ornstein. Paul C. Shields. "Universal Almost Sure Data Compression." Ann. Probab. 18 (2) 441 - 452, April, 1990.


Published: April, 1990
First available in Project Euclid: 19 April 2007

zbMATH: 0699.94004
MathSciNet: MR1055415
Digital Object Identifier: 10.1214/aop/1176990840

Primary: 94A34
Secondary: 28D20

Keywords: Entropy , rate-distortion function , Universal data compression

Rights: Copyright © 1990 Institute of Mathematical Statistics

Vol.18 • No. 2 • April, 1990
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