Nonhomogeneous, continuous-time Markov chains defined by series of proportional intensity matrices (Q1122870)

Let S be a countable state space with transition matrices P(s,t), where \(P_{ij}(s,t)=P\{X(t)=j| X(s)=i\}\) (\(\forall)t,s\geq 0\) and \(i,j\in S\). Let Q(t) be the intensity matrix. The authors, defining ergodicity and irreducibility of continuous-time Markov chains, show that if the discrete time Markov chain \(P_ n=I+(1/a_ n)A_ n\) is irreducible and the corresponding \(h_ n(t)\) does not vanish, then the intensity matrix \(Q(t)=\sum^{\infty}_{n=1}\ln (t)A_ n\) is irreducible, too. If \(P_ n\) is ergodic and \(h_ n(t)\) is ``large'' enough, then the nonhomogeneous, continuous-time chain is ergodic. For an intensity matrix A and a function h with \(h(t)\| A\| \leq 1\), \(\forall t\), it is shown that \(Q(t)=\sum^{\infty}_{n=1}(h(t))^ nA^ n\) is an intensity matrix. If \(P=I+(1/a)A\) is ergodic and if \(\int^{\infty}_{s}h(u)du=\infty\), then Q(t) is ergodic.