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^ζ where G is a “generator” matrix, that is G(i, j) > 0 for i, j distinct, and G(i, i) = −∑k≠i G(i, k), and ζ > 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 < ζ < 1, ζ > 1 or ζ = 1, converges in probability to a unique “stationary” vector μG, converges in law to a nontrivial mixture of point measures, or converges in law to a distribution μG with no atoms and full support on a simplex respectively, as n ↑ ∞. This last type of limit can be interpreted as a sort of “spreading” between the cases < ζ < 1 and ζ > 1 In particular, when G is appropriately chosen, μG is a Dirichlet distribution, reminiscent of results in Pólya urns.