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

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.