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29 March 2024
 
  » arxiv » math/0701798

 Article overview


Occupation laws for some time-nonhomogeneous Markov chains
Zach Dietz ; Sunder Sethuraman ;
Date 29 Jan 2007
Subject Probability
AbstractWe consider finite-state time-nonhomogeneous Markov chains where the probability of moving from state $i$ to state $j eq i$ at time $n$ is $G(i,j)/n^zeta$ for a ``generator’’ matrix $G$ and strength parameter $zeta>0$. 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 behaviors. These chains, however, exhibit some different, perhaps unexpected, asymptotic occupation laws depending on parameters.
Although on the one hand it is shown that the asymptotic position converges to a point-mixture for all $zeta>0$, on the other hand, the average position, when variously $0<zeta<1$, $zeta>1$ or $zeta=1$, is shown to converges to a constant, a point-mixture, or a distribution $mu_G$ with no atoms and full support on a certain simplex respectively. The last type of limit can be seen 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 with certain parameters, reminiscent of results in Polya urns.
Source arXiv, math/0701798
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