On a Theorem of Dubins and Freedman

Rabi Bhattacharya, Mukul Majumdar

Research output: Contribution to journalArticle

15 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)1067-1087
Number of pages21
JournalJournal of Theoretical Probability
Volume12
Issue number4
DOIs
StatePublished - Jan 1 1999

Keywords

  • Asymptotic stationarity
  • Iteration of i.i.d. maps
  • Markov processes
  • Monotone maps
  • Quadratic maps

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty

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