A Limit Theorem for a Class of Inhomogeneous Markov Processes

Let (t): t ∈ R⁺ or I⁺\ be an (aperiodic) irreducible Markov process with a finite state space S and transition rate qᵢⱼ(t) = p(i, j)(λ(t))U(i, j), where 0 ≤ U(i, j) ≤ ∞ and λ(t) is some suitable rate function with limt → ∞λ(t) = 0. We shall show in this article that there are constants h(i) ≥ 0 and βᵢ > 0 such that independent of X(0), limt → ∞P(X(t) = i) ÷ (λ(t))ʰ⁽ⁱ⁾ = βᵢ for each i ∈ S. The height function h is determined by $(p(i, j))$ and $(U(i, j))$. In particular, a limit distribution exists and concentrates on S = ∈ S: h(i) = 0\.