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On the Embedding Problem for Discrete-Time Markov Chains

Part of: Special matrices Markov processes Basic linear algebra

Published online by Cambridge University Press:  30 January 2018

Marie-Anne Guerry
Marie-Anne Guerry*
Affiliation:
Vrije Universiteit Brussel
*
∗ Postal address: Department of Business Technology and Operations, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium. Email address: marie-anne.guerry@vub.ac.be
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Abstract

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When a discrete-time homogenous Markov chain is observed at time intervals that correspond to its time unit, then the transition probabilities of the chain can be estimated using known maximum likelihood estimators. In this paper we consider a situation when a Markov chain is observed on time intervals with length equal to twice the time unit of the Markov chain. The issue then arises of characterizing probability matrices whose square root(s) are also probability matrices. This characterization is referred to in the literature as the embedding problem for discrete time Markov chains. The probability matrix which has probability root(s) is called embeddable.

In this paper for two-state Markov chains, necessary and sufficient conditions for embeddability are formulated and the probability square roots of the transition matrix are presented in analytic form. In finding conditions for the existence of probability square roots for (k x k) transition matrices, properties of row-normalized matrices are examined. Besides the existence of probability square roots, the uniqueness of these solutions is discussed: In the case of nonuniqueness, a procedure is introduced to identify a transition matrix that takes into account the specificity of the concrete context. In the case of nonexistence of a probability root, the concept of an approximate probability root is introduced as a solution of an optimization problem related to approximate nonnegative matrix factorization.

Keywords

Markov chainprobability matrixroot of a matrixembedding problem

MSC classification

Primary: 15A23: Factorization of matrices 15B51: Stochastic matrices 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 918 - 930
Copyright
© Applied Probability Trust 

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