An iterated random function with Lipschitz number one

A. Abrams*, H. Landau, Z. Landau, J. Pommersheim, E. Zaslow

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Consider the set of functions fθ(x) = |θ - x| on R. Define a, Markov process that starts with a point x0 ∈ R. and continues with Xk+1 = fθk+1 (xk) with each θk+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 πμ and any distribution converges under iteration to πμ (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 languageEnglish (US)
Pages (from-to)190-201
Number of pages12
JournalTheory of Probability and its Applications
Volume47
Issue number2
DOIs
StatePublished - Jun 26 2003

Keywords

  • Iterated random function
  • Markov process
  • Stationary distribution

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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