The embedding problem for Markov chains with three states

Abstract

In this paper we consider the embedding problem for Markov chains with three states. A non-singular stochastic matrix P is called embeddable if there exists a two-parameter family of stochastic matricessatisfyingand such thatThough extensive characterizations of embeddable n × n stochastic matrices have been given in (l), (2), (3), (6), and further characterizations of embeddable 3 × 3 stochastic matrices in (4), they do not provide, except in the case of 2 × 2 stochastic matrices, easily applicable necessary and sufficient conditions for embeddability.