The combinational structure of non-homogeneous Markov chains with countable states

Let P ( s , t ) denote a non-homogeneous continuous parameter Markov chain with countable state space E and parameter space [ a , b ] , − ∞ < a < b < ∞ . Let R ( s , t ) = { ( i , j ) : P i j ( s , t ) > 0 } . It is shown in this paper that R ( s , t ) is reflexive, transitive, and independent of ( s , t ) , s < t , if a certain weak homogeneity condition holds. It is also shown that the relation R ( s , t ) , unlike in the finite state space case, cannot be expressed even as an infinite (countable) product of reflexive transitive relations for certain non-homogeneous chains in the case when E is infinite.