Limit results on pattern entropy

We determine the entropy rate of patterns of certain random processes including all finite-entropy stationary processes. For independent and identically distributed (i.i.d.) processes, we also bound the speed at which the per-symbol pattern entropy converges to this rate, and show that patterns satisfy an asymptotic equipartition property. To derive some of these results we upper bound the probability that the nth variable in a random process differs from all preceding ones.