## Occupation laws for some time-nonhomogeneous Markov chains.(English)Zbl 1127.60068

Summary: 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 $$G(i,i)=-\sum_{k\neq i}G(i,k)$$, 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 $$0<\zeta<1$$, $$\zeta> 1$$ of $$\zeta=1$$, converges in probability to a unique stationary vector $$\nu_G$$, 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 $$n\uparrow\infty$$. 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 Pólya urns.

### MSC:

 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60F10 Large deviations
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