Electronic Journal of Probability

Occupation laws for some time-nonhomogeneous Markov chains

Zach Dietz and Sunder Sethuraman

Full-text: Open access

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.

Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 23, 661-683.

Dates
Accepted: 16 May 2007
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464818494

Digital Object Identifier
doi:10.1214/EJP.v12-413

Mathematical Reviews number (MathSciNet)
MR2318406

Zentralblatt MATH identifier
1127.60068

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60F10: Large deviations

Keywords
laws of large numbers nonhomogeneous Markov occupation reinforcement Dirichlet distribution

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Dietz, Zach; Sethuraman, Sunder. Occupation laws for some time-nonhomogeneous Markov chains. Electron. J. Probab. 12 (2007), paper no. 23, 661--683. doi:10.1214/EJP.v12-413. https://projecteuclid.org/euclid.ejp/1464818494


Export citation

References

  • Arratia, Richard. On the central role of scale invariant Poisson processes on
  • Arratia, Richard; Barbour, A. D.; Tavare, Simon. On Poisson-Dirichlet limits for random decomposable combinatorial structures. Combin. Probab. Comput. 8 (1999), no. 3, 193–208.
  • Arratia, Richard; Barbour, A. D.; Tavare, Simon. Logarithmic combinatorial structures: a probabilistic approach. EMS Monographs in Mathematics. European Mathematical Society (EMS), Z¸rich, (2003).
  • Athreya, Krishna B. On a characteristic property of Polya's urn. Studia Sci. Math. Hungar. 4 (1969) 31–35.
  • Bremaud, Pierre. Markov chains. Gibbs fields, Monte Carlo simulation, and queues. Texts in Applied Mathematics, 31. Springer-Verlag, New York, (1999).
  • Dobrushin, R. Limit theorems for Markov chains with two states. (Russian) Izv. Adad. Nauk SSSR 17:4 (1953), 291-330.
  • Gantert, Nina. Laws of large numbers for the annealing algorithm. Stochastic Process. Appl. 35 (1990), no. 2, 309–313.
  • Gouet, Raul. Strong convergence of proportions in a multicolor Polya urn. J. Appl. Probab. 34 (1997), no. 2, 426–435.
  • Hanen, Albert. Theoremes limites pour une suite de chaines de Markov. (French) Ann. Inst. H. Poincare 18 (1963) 197–301.
  • Hannig, Jan; Chong, Edwin K. P.; Kulkarni, Sanjeev R. Relative frequencies of generalized simulated annealing. Math. Oper. Res. 31 (2006), no. 1, 199–216.
  • Hartfiel, D. J. Dense sets of diagonalizable matrices. Proc. Amer. Math. Soc. 123 (1995), no. 6, 1669–1672.
  • Horn, Roger A.; Johnson, Charles R. Matrix analysis. Corrected reprint of the 1985 original. Cambridge University Press, Cambridge, (1990).
  • Isaacson, Dean L.; Madsen, Richard W. Markov chains. Theory and Applications. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York-London-Sydney, (1976).
  • Iosifescu, Marius. Finite Markov processes and their applications. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Ltd., Chichester; Editura Tehnicu a, Bucharest, (1980).
  • Iosifescu, M.; Theodorescu, R. Random processes and learning. Die Grundlehren der mathematischen Wissenschaften, Band 150. Springer-Verlag, New York, (1969).
  • Kotz, Samuel; Balakrishnan, N. Advances in urn models during the past two decades. Advances in combinatorial methods and applications to probability and statistics, 203–257, Stat. Ind. Technol., Birkhauser Boston, Boston, MA, (1997).
  • Kotz, Samuel; Balakrishnan, N.; Johnson, Norman L. Continuous multivariate distributions. Vol. 1. Models and applications. Second edition. Wiley Series in Probability and Statistics: Applied Probability and Statistics. Wiley-Interscience, New York, (20000.
  • Liu, Wen; Liu, Guo Xin. A class of strong laws for functionals of countable nonhomogeneous Markov chains. Statist. Probab. Lett. 22 (1995), no. 2, 87–96.
  • Miclo, Laurent. Sur les temps d'occupations des processus de Markov finis inhomogËnes a basse temperature. (French) [Occupation times of low-temperature nonhomogeneous finite Markov processes] Stochastics Stochastics Rep. 63 (1998), no. 1-2, 65–137.
  • Del Moral, P.; Miclo, L. Self-interacting Markov chains. Stoch. Anal. Appl. 24 (2006), no. 3, 615–660.
  • Pemantle, Robin. A survey of random processes with reinforcement. Probab. Surv. 4 (2007), 1–79 (electronic).
  • Pitman, Jim. Some developments of the Blackwell-MacQueen urn scheme. Statistics, probability and game theory, 245–267, IMS Lecture Notes Monogr. Ser., 30, Inst. Math. Statist., Hayward, CA, (1996).
  • Pitman, J. Combinatorial stochastic processes. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002. With a foreword by Jean Picard. Lecture Notes in Mathematics, 1875. Springer-Verlag, Berlin, (2006).
  • Sethuraman, Jayaram. A constructive definition of Dirichlet priors. Statist. Sinica 4 (1994), no. 2, 639–650.
  • Vervaat, W. Success epochs in Bernoulli trials (with applications in number theory). Mathematical Centre Tracts, 42. Mathematisch Centrum, Amsterdam, (1972).
  • Wen, Liu; Weiguo, Yang. An extension of Shannon-McMillan theorem and some limit properties for nonhomogeneous Markov chains. Stochastic Process. Appl. 61 (1996), no. 1, 129–145.
  • Winkler, Gerhard. Image analysis, random fields and Markov chain Monte Carlo methods. A mathematical introduction. Second edition. Applications of Mathematics 27. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, (2003).