## Abstract

Consider the set of functions f_{θ}(x) = |θ - x| on R. Define a, Markov process that starts with a point x_{0} ∈ R. and continues with X_{k+1} = f_{θk+1} (x_{k}) with each θ_{k+1} chosen from a fixed bounded distribution μ on R^{+}. We prove the conjecture of Letac that if μ is not supported on a lattice, then this process has a unique stationary distribution π_{μ} and any distribution converges under iteration to π_{μ} (in the weak-* topology). We also give a bound on the rate of convergence in the special case that μ is supported on a two-point set. We hope that the techniques will be useful for the study of other Markov processes where the transition functions have Lipschitz number one.

Original language | English (US) |
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Pages (from-to) | 190-201 |

Number of pages | 12 |

Journal | Theory of Probability and its Applications |

Volume | 47 |

Issue number | 2 |

DOIs | |

State | Published - 2003 |

## Keywords

- Iterated random function
- Markov process
- Stationary distribution

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty