Journal
LINEAR & MULTILINEAR ALGEBRA
Volume 62, Issue 3, Pages 362-385Publisher
TAYLOR & FRANCIS LTD
DOI: 10.1080/03081087.2013.777436
Keywords
limiting probability distribution vector; transition probability tensor; non-negative tensor; it Z-eigenvalue; iterative method; higher-order Markov chains
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Funding
- NNSF of China [10771075, 11271144]
- Centre for Mathematical Imaging and Vision, HKRGC [201812]
- HKBU FRG
- NSF of Guangdong province [91510631000021]
- research fund for the Doctoral program of higher education of China [20104407110001]
- Institute for Computational Mathematics and Centre for Mathematical Imaging and Vision, Hong Kong Baptist University
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In this paper, we propose and develop an iterative method to calculate a limiting probability distribution vector of a transition probability tensor arising from a higher order Markov chain. In the model, the computation of such limiting probability distribution vector can be formulated as a -eigenvalue problem associated with the eigenvalue 1 of where all the entries of are required to be non-negative and its summation must be equal to one. This is an analog of the matrix case for a limiting probability vector of a transition probability matrix arising from the first-order Markov chain. We show that if is a transition probability tensor, then solutions of this -eigenvalue problem exist. When is irreducible, all the entries of solutions are positie. With some suitable conditions of , the limiting probability distribution vector is even unique. Under the same uniqueness assumption, the linear convergence of the iterative method can be established. Numerical examples are presented to illustrate the theoretical results of the proposed model and the iterative method.
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