An iterated random function with Lipschitz number one
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918073648283648 |
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| author | Abrams, Aaron Landau, Henry Landau, Zeph Pommersheim, James Zaslow, Eric |
| author_facet | Abrams, Aaron Landau, Henry Landau, Zeph Pommersheim, James Zaslow, Eric |
| contents | Consider the set of functions $f_θ(x)=|θ-x|$ on $\mathbb{R}$. Define a Markov process that starts with a point $x_0 \in \mathbb{R}$ and continues with $x_{k+1}=f_{θ_{k+1}}(x_{k})$ with each $θ_{k+1}$ picked from a fixed bounded distribution $μ$ on $\mathbb{R}^+$. We prove the conjecture of G. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22420 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An iterated random function with Lipschitz number one Abrams, Aaron Landau, Henry Landau, Zeph Pommersheim, James Zaslow, Eric Probability Dynamical Systems 37A10 Consider the set of functions $f_θ(x)=|θ-x|$ on $\mathbb{R}$. Define a Markov process that starts with a point $x_0 \in \mathbb{R}$ and continues with $x_{k+1}=f_{θ_{k+1}}(x_{k})$ with each $θ_{k+1}$ picked from a fixed bounded distribution $μ$ on $\mathbb{R}^+$. We prove the conjecture of G. 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. |
| title | An iterated random function with Lipschitz number one |
| topic | Probability Dynamical Systems 37A10 |
| url | https://arxiv.org/abs/2506.22420 |