Moment-optimal finitary isomorphism for i.i.d. processes of equal entropy

TL;DR

This paper proves the existence of explicit finitary isomorphisms between i.i.d. processes of equal entropy with coding radii having finite moments for all t in (0,1/2), improving understanding of the isomorphism's finiteness properties.

Contribution

It establishes the first explicit construction of isomorphisms with finite t-moments for all t in (0,1/2) between i.i.d. processes of equal entropy, resolving a longstanding open problem.

Findings

Existence of isomorphisms with finite t-moments for all t in (0,1/2)

Explicit construction of such isomorphisms

Optimal tail behavior up to poly-logarithmic factors

Abstract

The finitary isomorphism theorem, due to Keane and Smorodinsky, raised the natural question of how "finite" the isomorphism can be, in terms of moments of the coding radius. More precisely, for which values does there exist an isomorphism between any two i.i.d. processes of equal entropy, with coding radii exhibiting finite t-moments? [3, 4]. Parry [13] and Krieger [10] showed that those finite moments must be lesser than 1 in general, and Harvey and Peres [5] showed that they must be lesser than 1/2 in general. However, the question for the range between 0 and 1/2 remained open, and in fact no general construction of an isomorphism was shown to exhibit any non trivial finite moments. In the present work we settle this problem, showing that between any two aperiodic Markov processes (and i.i.d. processes in particular) of the same entropy, there exists an isomorphism f with coding radii exhibiting finite t-moments for all t in (0,1/2). The isomorphism is constructed explicitly, and the tails of the radii are shown to be optimal up to a poly-logarithmic factor.