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<SUP><FONT FACE="Symbol">z</FONT></SUP> where G is a ``generator'' matrix, that is G(i,j)>0 for i,j distinct, and G(i,i)= -∑<SUB>k<FONT FACE="Symbol">≠</FONT>i</SUB> G(i,k), and <FONT FACE="Symbol">z</FONT>>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<<FONT FACE="Symbol">z</FONT><1, <FONT FACE="Symbol">z</FONT>>1 or <FONT FACE="Symbol">z</FONT>=1, converges in probability to a unique ``stationary'' vector <FONT FACE="Symbol">n</FONT><SUB>G</SUB>, converges in law to a nontrivial mixture of point measures, or converges in law to a distribution <FONT FACE="Symbol">m</FONT><SUB>G</SUB> with no atoms and full support on a simplex respectively, as n tends to infinity. This last type of limit can be interpreted as a sort of ``spreading'' between the cases 0<<FONT FACE="Symbol">z</FONT><1 and <FONT FACE="Symbol">z</FONT>>1. In particular, when G is appropriately chosen, <FONT FACE="Symbol">m</FONT><SUB>G</SUB> is a Dirichlet distribution, reminiscent of results in Polya urns.