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Small sets and Markov transition densities
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Kendall, Wilfrid S. and Montana, Giovanni. (2002) Small sets and Markov transition densities. Stochastic Processes and their Applications, Vol.99 (No.2). pp. 177194. ISSN 03044149

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Official URL: http://dx.doi.org/10.1016/S03044149(02)00090X
Abstract
The theory of general statespace Markov chains can be strongly related to the case of discrete statespace 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 statespaces with countably generated sigmaalgebras, though the minorization provided by the theory concerns small sets of order n and nstep 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 2step kernel density can be expressed as a countable sum of nonnegative 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 statespace Markov chain struggling to escape. We conclude by discussing complements to these results, including their relevance to Harrisrecurrent Markov chains and we relate the counterexample to Turan problems for bipartite graphs.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Statistics 
Library of Congress Subject Headings (LCSH):  Markov processes, Statespace methods 
Journal or Publication Title:  Stochastic Processes and their Applications 
Publisher:  Elsevier Science BV 
ISSN:  03044149 
Official Date:  June 2002 
Volume:  Vol.99 
Number:  No.2 
Number of Pages:  18 
Page Range:  pp. 177194 
Identification Number:  10.1016/S03044149(02)00090X 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Restricted or Subscription Access 
Funder:  Engineering and Physical Sciences Research Council (EPSRC), European Union (EU) 
Grant number:  GR/L5683 (EPSRC), ERBFMRXCT960095 (EU) 
References:  [1] K.B. Athreya and P. Ney. A new approach to the limit theory of recurrent Markov 
URI:  http://wrap.warwick.ac.uk/id/eprint/10903 
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