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Proceedings of the London Mathematical Society Advance Access published online on December 12, 2006
Proceedings of the London Mathematical Society, 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 dromik{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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