Universal finitary codes with exponential tails

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.