Skip to main content
Log in

Limit theorems for monotone Markov processes

  • Published:
Sankhya A Aims and scope Submit manuscript

Abstract

This article considers the convergence to steady states of Markov processes generated by the action of successive i.i.d. monotone maps on a subset S of an Eucledian space. Without requiring irreducibility or Harris recurrence, a “splitting” condition guarantees the existence of a unique invariant probability as well as an exponential rate of convergence to it in an appropriate metric. For a special class of Harris recurrent processes on [0,∞) of interest in economics, environmental studies and queuing theory, criteria are derived for polynomial and exponential rates of convergence to equilibrium in total variation distance. Central limit theorems follow as consequences.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bhattacharya, R.N. (1982). On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Z. Wahrsch. Verw. Gebiete, 60, 185–201.

    Article  MATH  MathSciNet  Google Scholar 

  • Bhattacharya, R.N. and Lee, O. (1988). Asymptotics of a class of Markov processes which are not in general irreducible. Ann. Probab., 16, 1333–1347 (correction (1997): Ann. Probab., 25, 1541–43).

    Article  MATH  MathSciNet  Google Scholar 

  • Bhattacharya, R.N. and Majumdar, M. (1999). On a theorem of Dubins and Freedman. J. Theoret. Probab., 12, 1067–1087.

    Article  MATH  MathSciNet  Google Scholar 

  • Bhattacharya, R.N. and Majumdar, M. (2001). On a class of random dynamical systems: theory and applications. J. Econom. Theory, 96, 208–229.

    Article  MATH  MathSciNet  Google Scholar 

  • Bhattacharya, R.N. and Majumdar, M. (2004). Stability in distribution of randomly perturbed quadratic maps as Markov processes. Ann. Appl. Probab., 14, 1802–1809.

    Article  MATH  MathSciNet  Google Scholar 

  • Bhattacharya, R.N. and Majumdar, M. (2007). Random Dynamical Systems: Theory and Applications. Cambridge University Press, Cambridge.

    Book  MATH  Google Scholar 

  • Bhattacharya, R.N. and Majumdar, M. (2010). Random iterates of monotone maps. Rev. Econ. Des., 14, 185–192.

    MATH  MathSciNet  Google Scholar 

  • Bhattacharya, R.N. and Ranga Rao, R. (1976). Normal Approximation and Asymptotic Expansions. John Wiley and Sons, New York.

    MATH  Google Scholar 

  • Bhattacharya, R.N. and Rao, B.V. (1993). Random iteration of two quadratic maps. In Stochastic Processes: A Festschrift in Honour of Gopinath Kallianpur, (Cambanis, S., Ghosh, J.K., Karandikar, R.L. and Sen, P.K., eds.). Springer-Verlag, New York, 13–22.

    Google Scholar 

  • Bhattacharya, R.N. and Waymire, E.C. (2009). Stochastic Processes with Applications. SIAM Classics in Applied Mathematics, 61. SIAM, Philadelphia.

    Google Scholar 

  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.

    MATH  Google Scholar 

  • Blackwell, D. (1965). Discounted dynamic programming. Ann. Math. Statist., 36, 226–235.

    Article  MATH  MathSciNet  Google Scholar 

  • Blumenthal, R.M. and Corson, H. (1972). On continuous collections of measures. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, 2, (L.M. Le Cam, J. Neyman and E.L. Scott, eds.). Univ. California Press, Berkeley, 33–40.

    Google Scholar 

  • Chakraborty, S. and Rao, B.V. (1998) Completeness of Bhattacharya metric on the space of probabilities. Statist. Probab. Lett., 36, 321–326.

    Article  MATH  MathSciNet  Google Scholar 

  • Diaconis, P. and Freedman, D. (1999) Iterated random functions. SIAM Rev., 41, 45–76.

    Article  MATH  MathSciNet  Google Scholar 

  • Diaconis, P. and Freedman, D. (1966). Invariant probabilities for certain Markov processes. Ann. Math. Statist., 37, 837–868.

    Article  MathSciNet  Google Scholar 

  • Ellner, S. (1984). Asymptotic behavior of some stochastic difference equation population models. J. Math. Biol., 19, 169–200.

    Article  MathSciNet  Google Scholar 

  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2. Second Edition. John Wiley and Sons, New York.

    Google Scholar 

  • Gordin, M.I. and Lifsic, B.A. (1978). The central limit theorem for stationary Markov processes (English translation). Soviet Math. Dokl., 19, 392–394.

    MATH  Google Scholar 

  • Hopenhayn, H.A. and Prescott, E.C. (1992). Stochastic monotonicity and stationary distributions for dynamic economies. Econometrica, 60, 1387–1406.

    Article  MATH  MathSciNet  Google Scholar 

  • Iams, S. and Majumdar, M. (2010). Stochastic equlibrium: concepts and computations for Lindley processes. Internat. J. of Econom. Theory, 6, 47–56.

    Article  Google Scholar 

  • Kifer, Y. (1986). Ergodic Theory of Random Transformations. Birkhauser, Boston.

    MATH  Google Scholar 

  • Lindley, D.V. (1952). The theory of queues with a single server. Math. Proc. Cambridge Philos. Soc., 48, 277.

    Article  MathSciNet  Google Scholar 

  • Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley and Sons, New York.

    MATH  Google Scholar 

  • Lund, R.B. and Tweedie, R.L. (1996). Geometric convergence rates for stochastically ordered Markov chains. Math. Oper. Res., 21, 182–194.

    Article  MATH  MathSciNet  Google Scholar 

  • Maitra, A. (1968). Discounted dynamic programming on compact metric spaces. Sankhyā, Ser. A, 27, 241–248.

    MathSciNet  Google Scholar 

  • Majumdar, M. and Mitra, T. (1983). Dynamic optimization with non-convex technology: The case of a linear objective function. Rev. Econom. Stud., 50, 143–151.

    Article  MATH  MathSciNet  Google Scholar 

  • Majumdar, M., Mitra, T. and Nyarko, Y. (1989). Dynamic optimization under uncertainty: non-convex feasible set. In Joan Robinson and Modern Economic Theory, (G. Feiwel et al., eds.). Macmillan, New York, 545–590.

    Google Scholar 

  • Meyn, S.P. and Tweedie, R.L. (1993). Markov Chains and Stochastic Stability. Springer-Verlag, New York.

    MATH  Google Scholar 

  • Ross, S.M. (1983). Introduction to Stochastic Dynamic Programming. Academic Press, New York.

    MATH  Google Scholar 

  • Spitzer, F. (1956). A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc., 82, 323–339.

    MATH  MathSciNet  Google Scholar 

  • Yahav, J.A. (1975). On a fixed point theorem and its stochastic equivalent. J. Appl. Probab., 12, 605–611.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rabi Bhattacharya.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bhattacharya, R., Majumdar, M. & Hashimzade, N. Limit theorems for monotone Markov processes. Sankhya 72, 170–190 (2010). https://doi.org/10.1007/s13171-010-0005-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13171-010-0005-6

AMS (2000) subject classification

Keywords and phrases

Navigation