Linear dynamics for the state vector of Markov chain functions
Keyword(s):
Let (φ(X n )) n be a function of a finite-state Markov chain (X n ) n . In this article, we investigate the conditions under which the random variables φ( n ) have the same distribution as Y n (for every n), where (Y n ) n is a Markov chain with fixed transition probability matrix. In other words, for a deterministic function φ, we investigate the conditions under which (X n ) n is weakly lumpable for the state vector. We show that the set of all probability distributions of X 0, such that (X n ) n is weakly lumpable for the state vector, can be finitely generated. The connections between our definition of lumpability and the usual one (i.e. as the proportional dynamics property) are discussed.
2004 ◽
Vol 36
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pp. 1198-1211
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1989 ◽
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pp. 757-766
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pp. 197-222
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pp. 159-166
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pp. 267-285
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pp. 623-629
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2018 ◽
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pp. 1552-1563
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