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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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