The evolution of aggregated Markov chains

For a stationary two-sided Markov chain (Xn)n∈Z with finite state-space I and a partition I=⋃ν=0s-1Iν we consider the aggregated sequence defined by Yn=ν if Xn∈Iν, which is also stationary but in general not Markovian. We present a tractable way to determine the transition probabilities of (Yn)n∈Z, either given a finite part of its past or given its infinite past. These probabilities are linked to the Radon-Nikodym derivative of PUn|Xn=i with respect to PUn, where Un=∑m=1∞s-mYn-m.