Linear and Multilinear Algebra· 2013Q1
On the limiting probability distribution of a transition probability tensor
- 186citations
- Q1SCImago
- 2013year
Short summary
A novel iterative method is presented to compute the limiting probability distribution vector of a transition probability tensor in higher-order Markov chains, formulated as a specific eigenvalue problem.
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Key points
- Proposes an iterative method for calculating the limiting probability distribution vector of a transition probability tensor in higher-order Markov chains.
- The problem is framed as a $\lambda$-eigenvalue problem with $\lambda=1$ for a tensor with non-negative entries summing to one.
- Guarantees existence of solutions for transition probability tensors, with positive entries for irreducible tensors.
- Establishes linear convergence for the iterative method under uniqueness assumptions for the limiting vector.
AI-generated from the title and abstract; the full text is not read.
Abstract
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.
The authors' abstract, as published at the source. Linear and Multilinear Algebra, 2013 · DOI ↗
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Field: Computational Mathematics
Computational MathematicsMathematics