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Convergence of a transition probability tensor of a higher‐order Markov chain to the stationary probability vector
Author(s) -
Bozorgmanesh Hassan,
Hajarian Masoud
Publication year - 2016
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2063
Subject(s) - markov chain , mathematics , continuous time markov chain , tensor (intrinsic definition) , probability distribution , probability measure , stationary distribution , convergence (economics) , additive markov chain , stationary sequence , markov chain mixing time , markov model , statistical physics , markov property , mathematical analysis , random variable , pure mathematics , statistics , physics , economics , economic growth
Summary In this paper, first we introduce a new tensor product for a transition probability tensor originating from a higher‐order Markov chain. Subsequently, some properties of the new tensor product are explained, and its relationship with the stationary probability vector is studied. Also, similarity between results obtained by this new product and the first‐order case is shown. Furthermore, we prove the convergence of a transition probability tensor to the stationary probability vector. Finally, we show how to achieve a stationary probability vector with some numerical examples and make some comparison between the proposed method and another existing method for obtaining stationary probability vectors. Copyright © 2016 John Wiley & Sons, Ltd.