Limit theorem for Markov processes with finite number of states (Q804093)

Let \(X_{\epsilon}(t)\) for any \(\epsilon >0\) be a homogeneous Markov process with a finite set of possible states E and transition probability \(P^{\epsilon}_{ij}(t)\) for t. Let X(t) be the limiting process with transition probability \(P_{ij}(t)=\lim_{\epsilon \to 0}P^{\epsilon}_{ij}(t)\). The set of possible states consists of subsets \(E_ k\), \(k=1,...,r\), where \(E=E_ 1\cup...\cup E_ r,\) \(E_ k\cap E_ m=\emptyset\) and let us fix r states \(o_ 1\in E_ 1,...,o_ r\in E_ r\). Finally let \(f^{\epsilon}_ k(t)=\sum_{j\in A}P^{\epsilon}_{o_{kj}}(t)\), \(k=1,...,r\), where A is a subset of E. The paper investigates the asymptotic behaviour of \(f^{\epsilon}_ k(t/\epsilon)\) as \(\epsilon\to 0\), i.e. it describes the asymptotic behaviour of the solution of the multidimensional renewal equation for the corresponding class of homogeneous Markov processes.