Occupation laws for some time-nonhomogeneous Markov chains
@article{EJP413,
author = {Zach Dietz and Sunder Sethuraman},
title = {Occupation laws for some time-nonhomogeneous Markov chains},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {12},
year = {2007},
keywords = {laws of large numbers, nonhomogeneous, Markov, occupation, reinforcement, Dirichlet distribution.},
abstract = {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.
},
pages = {no. 23, 661-683},
issn = {1083-6489},
doi = {10.1214/EJP.v12-413},
url = {http://ejp.ejpecp.org/article/view/413}}