Markov Chains and Mappings of Distributions on Compact Spaces
David Aldous, Shi Feng · Latin American Journal of Probability and Mathematical Statistics · 2024
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).So 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?We present a few rigorous results, to complement our extensive simulation study elsewhere.