### 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 language | English (US) |
---|---|

Pages (from-to) | 1067-1087 |

Number of pages | 21 |

Journal | Journal of Theoretical Probability |

Volume | 12 |

Issue number | 4 |

State | Published - 1999 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

*Journal of Theoretical Probability*,

*12*(4), 1067-1087.

**On a Theorem of Dubins and Freedman.** / Bhattacharya, Rabindra N; Majumdar, Mukul.

Research output: Contribution to journal › Article

*Journal of Theoretical Probability*, vol. 12, no. 4, pp. 1067-1087.

}

TY - JOUR

T1 - On a Theorem of Dubins and Freedman

AU - Bhattacharya, Rabindra N

AU - Majumdar, Mukul

PY - 1999

Y1 - 1999

N2 - 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.

AB - 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.

KW - Asymptotic stationarity

KW - Iteration of i.i.d. maps

KW - Markov processes

KW - Monotone maps

KW - Quadratic maps

UR - http://www.scopus.com/inward/record.url?scp=0005408547&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0005408547&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0005408547

VL - 12

SP - 1067

EP - 1087

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

IS - 4

ER -