Abstract: The finitary isomorphism theorem, due to Keane and Smorodinsky, raised the natural question of how “finite” can the isomorphism be, in terms of moments of the coding radius. More precisely, for which values t>0 does there exist, for any two i.i.d. processes of equal entropy, an isomorphism with coding radii exhibiting finite t-moments? Parry and Krieger showed that the 1st moment is not finite in general, and Harvey and Peres showed that the 1/2-moment is not finite in general. However, the question for 0 < t < 1/2 remained open.
Our results complement the above results, showing that between any two i.i.d. processes of the same entropy, there exists an isomorphism with coding radii exhibiting finite t-moments for all 0 < t < 1/2. We will then generalize some of the known theory to processes over Z^d, d>1.
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