Occupation laws for some time-nonhomogeneous Markov chains

Zach Dietz, Sunder Sethuraman

Research output: Contribution to journalArticle

4 Scopus citations

Abstract

We consider finite-state time-nonhomogeneous Markov chains whose transition matrix at time n is I + G/nζ where G is a “generator” matrix, that is G(i, j) > 0 for i, j distinct, and G(i, i) = −∑k≠i G(i, k), and ζ > 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 < ζ < 1, ζ > 1 or ζ = 1, converges in probability to a unique “stationary” vector μG, converges in law to a nontrivial mixture of point measures, or converges in law to a distribution μG with no atoms and full support on a simplex respectively, as n ↑ ∞. This last type of limit can be interpreted as a sort of “spreading” between the cases < ζ < 1 and ζ > 1 In particular, when G is appropriately chosen, μG is a Dirichlet distribution, reminiscent of results in Pólya urns.

Original languageEnglish (US)
Pages (from-to)661-683
Number of pages23
JournalElectronic Journal of Probability
Volume12
DOIs
StatePublished - Jan 1 2007
Externally publishedYes

Keywords

  • Dirichlet distribution
  • Laws of large numbers
  • Markov
  • Nonhomogeneous
  • Occupation
  • Reinforcement

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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