On the general transformation of Markov chains
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Abstract
In this work, we consider a class of homogeneous irreducible discrete states Markov process , taking values in a partition of some topological space . Using a notion of fractional state, we introduce a general transformation such that remains a Markov process. This general transformation includes also lumpability. Infinite kinds of this expanded Markov chain can be constructed, however lumped Markov chains are finite or may not exist .
Using general transformation and fractional states, we give a necessary and sufficient condition under which the transformed process remains still a Markov chain
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