On the Entropy Rate of Pattern Processes

We study the entropy rate of pattern sequences of stochastic processes, and its relationship to the entropy rate of the original process. We give a complete characterization of this relationship for independent and identically distributed (i.i.d.) processes over arbitrary alphabets, stationary ergodic processes over discrete alphabets, and a broad family of stationary ergodic processes over uncountable alphabets. For cases where the entropy rate of the pattern process is infinite, we characterize the possible growth rate of the block entropy