The evolution of aggregated Markov chains

Abstract

For a stationary two-sided Markov chain (X-n)(n is an element of Z) with finite state-space I and a partition I = Uv=0s-1Iv we consider the aggregated sequence defined by Y-n = v if X-n is an element of I-v, which is also stationary but in general not Markovian. We present a tractable way to determine the transition probabilities of either given a finite part of its past or given its infinite past. These probabilities are linked to the Radon-Nikodym derivative of P-Un vertical bar X-n=i with respect to P-Un, where U-n = Sigma(infinity)(m=1)s(-m)Y(n-m). (c) 2005 Elsevier B.V. All rights reserved.