Occupation laws for some time-nonhomogeneous Markov chains

We consider finite-state time-nonhomogeneous Markov chains whose transition matrix at time n is I + G/n(zeta) where G is a "generator" matrix, that is G(i, j) > 0 for i, j distinct, and G(i, i) = - Sigma(k not equal i) G(i, k), and zeta > 0 is a strength parameter. In these chains, as time grows, the positions are less and less likely to change, and so form simple models of age-dependent time-reinforcing schemes. These chains, however, exhibit a trichotomy of occupation behaviors depending on parameters. We show that the average occupation or empirical distribution vector up to time n, when variously 0 < zeta < 1,zeta > 1 or zeta = 1, converges in probability to a unique "stationary" vector. G, converges in law to a nontrivial mixture of point measures, or converges in law to a distribution mu(G) with no atoms and full support on a simplex respectively, as n up arrow infinity. This last type of limit can be interpreted as a sort of "spreading" between the cases 0 < zeta < 1 and zeta > 1. In particular, when G is appropriately chosen, mu(G) is a Dirichlet distribution, reminiscent of results in Polya urns.