### Abstract

We consider finite-state time-nonhomogeneous Markov chains whose transition matrix at time n is I + G/n^{ζ} 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 V _{G}, converges in law to a nontrivial mixture of point measures, or converges in law to a distribution μ_{G} 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 0 < ζ 1 and ζ > 1. In particular, when G is appropriately chosen, μ_{G} is a Dirichlet distribution, reminiscent of results in Pólya urns.

Original language | English (US) |
---|---|

Pages (from-to) | 661-683 |

Number of pages | 23 |

Journal | Electronic Journal of Probability |

Volume | 12 |

State | Published - May 15 2007 |

Externally published | Yes |

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### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Electronic Journal of Probability*,

*12*, 661-683.

**Occupation laws for some time-nonhomogeneous Markov chains.** / Dietz, Zach; Sethuraman, Sunder.

Research output: Contribution to journal › Article

*Electronic Journal of Probability*, vol. 12, pp. 661-683.

}

TY - JOUR

T1 - Occupation laws for some time-nonhomogeneous Markov chains

AU - Dietz, Zach

AU - Sethuraman, Sunder

PY - 2007/5/15

Y1 - 2007/5/15

N2 - 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 V G, 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 0 < ζ 1 and ζ > 1. In particular, when G is appropriately chosen, μG is a Dirichlet distribution, reminiscent of results in Pólya urns.

AB - 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 V G, 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 0 < ζ 1 and ζ > 1. In particular, when G is appropriately chosen, μG is a Dirichlet distribution, reminiscent of results in Pólya urns.

KW - Dirichlet distribution

KW - Laws of large numbers

KW - Markov

KW - Nonhomogeneous

KW - Occupation

KW - Reinforcement

UR - http://www.scopus.com/inward/record.url?scp=34249030618&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34249030618&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:34249030618

VL - 12

SP - 661

EP - 683

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

ER -