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arXiv · 2403.18153

Markov chains and mappings of distributions on compact spaces II: Numerics and Conjectures

David J. Aldous·Madelyn Cruz·Shi Feng
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Abstract

Consider a compact metric space $S$ and a pair $(j,k)$ with $k \ge 2$ and $1 \le j \le k$. For any probability distribution $θ\in P(S)$, define a Markov chain on $S$ by: from state $s$, take $k$ i.i.d. ($θ$) samples, and jump to the $j$'th closest. Such a chain converges in distribution to a unique stationary distribution, say $π_{j,k}(θ)$. This defines a mapping $π_{j,k}: P(S) \to P(S)$. What happens when we iterate this mapping? In particular, what are the fixed points of this mapping? A few results are proved in a companion article; this article, not intended for formal publication, records numerical studies and conjectures.

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David J. Aldous, Madelyn Cruz, Shi Feng. 2024-03-26. Markov chains and mappings of distributions on compact spaces II: Numerics and Conjectures. https://arxiv.org/abs/2403.18153

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