Moments of finitary factor maps between Bernoulli processes

Abstract

The problem of what moments can exist for the coding radius of a finitary map between two i.i.d. processes, has been extensively studied in the case of $\mathbb{Z}$-processes. Here we treat this problem for factor maps between ${\mathbb{Z}}^d$-processes ($d>1$). By modeling the homomorphism with a map between spaces of finite sequences, we extend Harvey and Peres' result, showing that for a finitary homomorphism between two i.i.d. processes of equal entropy, if the coding radius of the map has a finite $\frac{d}{2}$-moment, then the two processes share the same informational variance. We use our modeling technique to prove a "Schmidt-type theorem" - that in case the above homomorphism has a coding radius of exponential tails, then the two processes are essentially the same. This result appears to be new even for the one-dimensional case, addressing a question of Angel and Spinka.