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2007 Occupation laws for some time-nonhomogeneous Markov chains
Zach Dietz, Sunder Sethuraman
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Electron. J. Probab. 12: 661-683 (2007). DOI: 10.1214/EJP.v12-413


We consider finite-state time-nonhomogeneous Markov chains whose transition matrix at time $n$ is $I+G/n^z$ where $G$ is a ``generator'' matrix, that is $G(i,j)>0$ for $i,j$ distinct, and $G(i,i)= -\sum_{k\ne i} G(i,k)$, and $z>0$ 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 $0< z< 1$, $z>1$ or $z=1$, converges in probability to a unique ``stationary'' vector $n_G$, converges in law to a nontrivial mixture of point measures, or converges in law to a distribution $m_G$ 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 $0< z < 1$ and $z>1$.

In particular, when $G$ is appropriately chosen, $m_G$ is a Dirichlet distribution, reminiscent of results in Polya urns.


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Zach Dietz. Sunder Sethuraman. "Occupation laws for some time-nonhomogeneous Markov chains." Electron. J. Probab. 12 661 - 683, 2007.


Accepted: 16 May 2007; Published: 2007
First available in Project Euclid: 1 June 2016

zbMATH: 1127.60068
MathSciNet: MR2318406
Digital Object Identifier: 10.1214/EJP.v12-413

Primary: 60J10
Secondary: 60F10

Keywords: Dirichlet distribution , laws of large numbers , Markov , nonhomogeneous , occupation , Reinforcement

Vol.12 • 2007
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