Nonhomogeneous, continuous-time Markov chains defined by series of proportional intensity matrices

Let A1, A2,..., be commuting intensity matrices of homogeneous, continuous-time Markov chains. The irreducibility and ergodicity of nohomogeneous, continuous-time Markov chains defined by intensity matrices of the form Q(t) = [summation operator] hn(t)An, hn(t) [greater-or-equal, slanted]0, are studied in terms of corresponding discrete-time chains. By defining transition matrices of homogenous, discrete-time chains as it is found that if one Pn is irreducible and the cor does not vanish then Q(t) is irreducible. Similarly, if one of the Pn's (or the average of a finite number of the Pn's) is ergodic and the corresponding hn(t) is large enough ([integral operator][infinity]s hn(t)du=[infinity]) then the nonhomogeneous, continuous-time chain is ergodic. For an intensity matrix A and a nonnegative function h(t) with h(t)||A|| < 1 for all t, it is shown that Q(t) = [summation operator][infinity]n=1 (h(t))nAn is an intensity matrix. Moreover, if is ergodic and if [integral operator][infinity]s h(u) DU = [infinity], then Qergodic.