On the Imbedding Problem for Three-state Time Homogeneous Markov Chains with Coinciding Negative Eigenvalues

TL;DR

This paper establishes a new necessary and sufficient condition for the imbedding problem of certain 3x3 Markov chains with negative eigenvalues, avoiding complex matrix computations, and provides an implicit characterization related to prior work.

Contribution

It introduces a novel parameterization approach to solve the imbedding problem for 3x3 Markov chains with negative eigenvalues, simplifying the analysis.

Findings

Derived a new condition for the imbedding problem

Avoided matrix logarithm and square root calculations

Connected to and extended previous implicit descriptions

Abstract

For an indecomposable $3\times 3$ stochastic matrix (i.e., 1-step transition probability matrix) with coinciding negative eigenvalues, a new necessary and sufficient condition of the imbedding problem for time homogeneous Markov chains is shown by means of an alternate parameterization of the transition rate matrix (i.e., intensity matrix, infinitesimal generator), which avoids calculating matrix logarithm or matrix square root. In addition, an implicit description of the imbedding problem for the $3\times 3$ stochastic matrix in Johansen [J. Lond. Math. Soc., 8, 345-351. (1974)] is pointed out.