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

2024/03/26 by David Aldous, Aldous, David J., Madelyn Cruz +3
Mathematics · #60J05 #FOS: Mathematics #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2403.18153

openalex publication_date 2024/03/26 · openalex created_date 2024/03/29 · openalex updated_date 2026/07/28

Abstract

Consider a compact metric space S and a pair (j,k) with k ≥ 2 and 1 ≤ j ≤ k. For any probability distribution θ∈ 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) → 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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