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/nζ where G is a “generator” matrix, that is G(i, j) > 0 for i, j distinct, and G(i, i) = − ∑ k 6=iG(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 some different, perhaps unexpected, occupation behaviors depending on parameters. Although it is shown, on the one hand, that the position at time n converges to a point-mixture for all ζ > 0, on the other hand, the average occupation vector up to time n, when variously 0 < ζ < 1, ζ > 1 or ζ = 1, is seen to converge to a constant, a pointmixture, or 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 0 < ζ < 1 and ζ > 1. In particular, when G is appropriately chosen, intriguingly, μG is a Dirichlet distribution, reminiscent of results in Pólya urns. Research supported in part by NSA-H982300510041 and NSF-DMS-0504193