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Proceedings of the London Mathematical Society Advance Access originally published online on December 12, 2006
Proceedings of the London Mathematical Society 2007 94(2):475-496; doi:10.1112/plms/pdl018
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© 2006 London Mathematical Society

Universal finitary codes with exponential tails

Nate Harvey1, Alexander E. Holroyd2, Yuval Peres3 and Dan Romik4

1 Department of Mathematics
University of California at Berkeley
Berkeley, CA 94720
USA
2 Department of Mathematics
University of British Columbia
Vancouver, BC V6T 1Z2
Canada
holroyd{at}math.ubc.ca
3 Departments of Statistics and Mathematics
University of California at Berkeley
CA 94720, Berkeley
USA
peres{at}stat.berkeley.edu
4 The Mathematical Sciences Research Institute
17 Gauss Way Berkeley, CA 94720-5070
USA
romik{at}msri.org

Received 23 February 2005. Revision received 10 May 2006.

In 1977, Keane and Smorodinsky showed that there exists a finitary homomorphism from any finite-alphabet Bernoulli process to any other finite-alphabet Bernoulli process of strictly lower entropy. In 1996, Serafin proved the existence of a finitary homomorphism with finite expected coding length. In this paper, we construct such a homomorphism in which the coding length has exponential tails. Our construction is source-universal, in the sense that it does not use any information on the source distribution other than the alphabet size and a bound on the entropy gap between the source and target distributions. We also indicate how our methods can be extended to prove a source-specific version of the result for Markov chains.


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