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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Athreya, K. B. (1975). Stochastic iteration of stable processes. In Puri, M. L., (ed.), Stochastic Processes and Related Topics, Academic Press, New York.
Athreya, K. B., and Dai, J. B. (1998). Invariant measures for i.i.d. iteration of logistic maps. Iowa University, Department of Mathematics. Technical Reprint.
Barnsley, M. E., and Demko, S. (1985). Iterated function systems and the global construction of fractals. Proc. Royal Soc. London A 399, 243–275.
Barnsley, M. E., and Elton, J. H. (1988). A new class of Markov processes for image coding. Adv. Appl. Prob. 20, 14–32.
Bhattacharya, R. N., and Goswami, A. (1999). A Markovian algorithm for random continued fractions and a computation of their singular equilibria. In Basu, A. K., Ghosh, J. K., Sen, P. K., and Sinha, B. K. (eds.), Perspectives in Statistical Science, Proc. Third Calcutta Triennial Symp., Oxford Univ. Press, New Delhi. (in press).
Bhattacharya, R. N., and Lee, O. (1988). Asymptotics of a class of Markov processes which are not in general irreducible. Ann. Prob. 16, 1333–1347. [Correction, ibid. (1997), 25, 1541–1543].
Bhattacharya, R. N., and Majumdar, M. (1999). Convergence to equilibrium of random dynamical systems generated by i.i.d. monotone maps, with applications to economics. In Ghosh, S. (ed.), Asymptotics, Nonparametrics and Time Series: A Festschrift for M. L. Puri, Marcel Dekker, New York.
Bhattacharya, R. N., and Rao, B. V. (1993). Random iteration of two quadratic maps. In Cambanis, S., Gosh, J. K., Karandikar, R. L., and Sen, P. K. (eds.), Stochastic Processes: A Festschrift in Honor of Gropinath Kallianpur, Springer Verlag, New York.
Bhattacharya, R. N., and Waymire, E. C. (1990). Stochastic Processes with Applications, Wiley, New York.
Billingsley, P. (1968). Convergence of Probability Measures, Wiley, New York.
Breiman, L. (1968). Probability, Reading, Massachusetts.
Chakraborty, S., and Rao, B. V. (1998). Completeness of the Bhattacharya metric in the space of probabilities. Stat. Prob. Lett. 36, 321–326.
Chamayou, J. F., and Letac, G. (1991). Explicit stationary distributions for compositions of random functions and products of random matrices. J. Theor. Prob. 4, 3–36.
Chassiang, P., Letac, G., and Mora, M. (1984). Brocot sequence and random walks in SL(2, ℝ). Probability Measures on Groups VII. Lecture Notes in Math., Springer, New York.
Devaney, R. L. (1989). An Introduction to Chaotic Dynamical Systems, Second Edition, Addison-Wesley, New York.
Doob, J. L. (1953). Stochastic Processes. Wiley, New York.
Dubins, L. E., and Freedman, D. A. (1966). Invariant probabilities for certain Markov Processes. Ann. Math. Stat. 37, 837–848.
Friedman, A. (1982). Foundations of Modern Analysis, Dover, New York.
Kifer, Yu (1986). Ergodic Theory of Random Transformations. Birkhauser, Boston.
Letac, G., and Sheshadri, V. (1983). A characterization of the generalized inverse Gaussian distribution by continued fractions. Z. Wahrsch. verw. Geb. 62, 485–489.
Liggett, T. (Undated). Private communication.
Majumdar, M., Mitra, T., and Nyarko, Y. (1989). Dynamic optimization under uncertainty: Nonconvex feasible set. In Feiwel, G. R. (ed.), Joan Robinson and Modern Economic Theory, MacMillan, London, pp. 545–590.
Majumdar, M., and Radner, R. (1992). Survival under production uncertainty. In Majumdar, M. (ed.), Equilibrium and Dynamics, MacMillan, London.
Meyn, S. P., and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability, Springer-Verlag, New York.
Mirman, L. J. (1980). One sector economic growth under uncertainty: A survey. In Dempster, M. A. H., (ed.), Stochastic Programming, Academic Press, New York.
Sandefur, J. T. (1990). Discrete Dynamical Systems: Theory and Applications, Clarendon Press, Oxford.
Tong, H. (1990). Nonlinear Time Series: A Dynamical System Approach, Oxford University Press, Oxford.
Tweedie, R. L. (1975). Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space. Stoch. Proc. Appl. 3, 385–403.
Tweedie, R. L. (1983). Criteria for rates of convergence of Markov chains, with application to queuing and storage theory. In Kingman, J. F. C., and Reuter, G. E. H. (eds.), Papers in Probability, Statistics and Analysis, pp. 260–276, Cambridge University Press, Cambridge.
Yahav, J. A. (1975). On a fixed point theorem and its stochastic equivalent. J. Appl. Prob. 12, 605–611.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bhattacharya, R., Majumdar, M. On a Theorem of Dubins and Freedman. Journal of Theoretical Probability 12, 1067–1087 (1999). https://doi.org/10.1023/A:1021601421920
Issue Date:
DOI: https://doi.org/10.1023/A:1021601421920