Indexing metadata

Occupation laws for some time-nonhomogeneous Markov chains


 
Dublin Core PKP Metadata Items Metadata for this Document
 
1. Title Title of document Occupation laws for some time-nonhomogeneous Markov chains
 
2. Creator Author's name, affiliation, country Zach Dietz; Tulane University
 
2. Creator Author's name, affiliation, country Sunder Sethuraman; Iowa State University
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) laws of large numbers, nonhomogeneous, Markov, occupation, reinforcement, Dirichlet distribution.
 
3. Subject Subject classification Primary 60J10; secondary 60F10
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) NSF, NSA
 
7. Date (YYYY-MM-DD) 2007-05-16
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/413
 
10. Identifier Digital Object Identifier 10.1214/EJP.v12-413
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 12
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
15. Rights Copyright and permissions The Electronic Journal of Probability applies the Creative Commons Attribution License (CCAL) to all articles we publish in this journal. Under the CCAL, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles published in EJP, so long as the original authors and source are credited. This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available.

Summary of the Creative Commons Attribution License

You are free
  • to copy, distribute, display, and perform the work
  • to make derivative works
  • to make commercial use of the work
under the following condition of Attribution: others must attribute the work if displayed on the web or stored in any electronic archive by making a link back to the website of EJP via its Digital Object Identifier (DOI), or if published in other media by acknowledging prior publication in this Journal with a precise citation including the DOI. For any further reuse or distribution, the same terms apply. Any of these conditions can be waived by permission of the Corresponding Author.