An iterated random function with Lipschitz number one
Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 2, pp. 286-300
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Consider the set of functions $f_{\theta}(x)=|\theta -x|$ on $\mathbf R$. Define a Markov process that starts with a point $x_0 \in \mathbf R$ and continues with $x_{k+1}=f_{\theta_{k+1}}(x_{k})$ with each $\theta _{k+1}$ chosen from a fixed bounded distribution $\mu$ on ${\mathbf R}^+$. We prove the conjecture of Letac that if $\mu$ is not supported on a lattice, then this process has a unique stationary distribution $\pi_{\mu}$ and any distribution converges under iteration to $\pi_{\mu}$ (in the weak-$^*$ topology). We also give a bound on the rate of convergence in the special case that $\mu$ 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.
Keywords:
iterated random function, Markov process, stationary distribution.
@article{TVP_2002_47_2_a4,
author = {A. Abrams and H. Landau and Z. Landau and J. Pommersheim and E. Zaslow},
title = {An iterated random function with {Lipschitz} number one},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {286--300},
year = {2002},
volume = {47},
number = {2},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TVP_2002_47_2_a4/}
}
TY - JOUR AU - A. Abrams AU - H. Landau AU - Z. Landau AU - J. Pommersheim AU - E. Zaslow TI - An iterated random function with Lipschitz number one JO - Teoriâ veroâtnostej i ee primeneniâ PY - 2002 SP - 286 EP - 300 VL - 47 IS - 2 UR - http://geodesic.mathdoc.fr/item/TVP_2002_47_2_a4/ LA - en ID - TVP_2002_47_2_a4 ER -
A. Abrams; H. Landau; Z. Landau; J. Pommersheim; E. Zaslow. An iterated random function with Lipschitz number one. Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 2, pp. 286-300. http://geodesic.mathdoc.fr/item/TVP_2002_47_2_a4/