Small sets and Markov transition densities
AbstractThe theory of general state-space Markov chains can be strongly related to the case of discrete state-space by use of the notion of small sets and associated minorization conditions. The general theory shows that small sets exist for all Markov chains on state-spaces with countably generated [sigma]-algebras, though the minorization provided by the theory concerns small sets of order n and n-step transition kernels for some unspecified n. Partly motivated by the growing importance of small sets for Markov chain Monte Carlo and Coupling from the Past, we show that in general there need be no small sets of order n=1 even if the kernel is assumed to have a density function (though of course one can take n=1 if the kernel density is continuous). However, n=2 will suffice for kernels with densities (integral kernels), and in fact small sets of order 2 abound in the technical sense that the 2-step kernel density can be expressed as a countable sum of non-negative separable summands based on small sets. This can be exploited to produce a representation using a latent discrete Markov chain; indeed one might say, inside every Markov chain with measurable transition density there is a discrete state-space Markov chain struggling to escape. We conclude by discussing complements to these results, including their relevance to Harris-recurrent Markov chains and we relate the counterexample to Turán problems for bipartite graphs.
Download InfoIf you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.
Bibliographic InfoArticle provided by Elsevier in its journal Stochastic Processes and their Applications.
Volume (Year): 99 (2002)
Issue (Month): 2 (June)
Contact details of provider:
Web page: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description
You can help add them by filling out this form.
CitEc Project, subscribe to its RSS feed for this item.
- van Lieshout, M.N.M. & Stoica, R.S., 2006. "Perfect simulation for marked point processes," Computational Statistics & Data Analysis, Elsevier, vol. 51(2), pages 679-698, November.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Zhang, Lei).
If references are entirely missing, you can add them using this form.