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On a Theorem of Dubins and Freedman

Author

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  • Rabi Bhattacharya

    (Indiana University)

  • Mukul Majumdar

    (Cornell University)

Abstract

Under a notion of “splitting” the existence of a unique invariant probability, and a geometric rate of convergence to it in an appropriate metric, are established for Markov processes on a general state space S generated by iterations of i.i.d. maps on S. As corollaries we derive extensions of earlier results of Dubins and Freedman;(17) Yahav;(30) and Bhattacharya and Lee(6) for monotone maps. The general theorem applies in other contexts as well. It is also shown that the Dubins–Freedman result on the “necessity” of splitting in the case of increasing maps does not hold for decreasing maps, although the sufficiency part holds for both. In addition, the asymptotic stationarity of the process generated by i.i.d. nondecreasing maps is established without the requirement of continuity. Finally, the theory is applied to the random iteration of two (nonmonotone) quadratic maps each with two repelling fixed points and an attractive period-two orbit.

Suggested Citation

  • Rabi Bhattacharya & Mukul Majumdar, 1999. "On a Theorem of Dubins and Freedman," Journal of Theoretical Probability, Springer, vol. 12(4), pages 1067-1087, October.
    • Handle: RePEc:spr:jotpro:v:12:y:1999:i:4:d:10.1023_a:1021601421920
    DOI: 10.1023/A:1021601421920
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    References listed on IDEAS

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    1. Mukul Majumdar & Roy Radner, 1992. "Survival under Production Uncertainty," Palgrave Macmillan Books, in: Mukul Majumdar (ed.), Equilibrium and Dynamics, chapter 8, pages 179-200, Palgrave Macmillan.
    2. Chakraborty, Santanu & Rao, B. V., 1998. "Completeness of Bhattacharya metric on the space of probabilities," Statistics & Probability Letters, Elsevier, vol. 36(4), pages 321-326, January.
    3. Tweedie, Richard L., 1975. "Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space," Stochastic Processes and their Applications, Elsevier, vol. 3(4), pages 385-403, October.
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