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

David J. Aldous, Madelyn Cruz, Shi Feng

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.

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

Abstract

Consider a compact metric space and a pair with and . For any probability distribution , define a Markov chain on by: from state , take i.i.d. () samples, and jump to the 'th closest. Such a chain converges in distribution to a unique stationary distribution, say . This defines a mapping . 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.
Paper Structure (26 sections, 8 theorems, 41 equations, 17 figures, 2 tables)

This paper contains 26 sections, 8 theorems, 41 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Formulating a program
  3. Toward a big picture
  4. Outline of this article
  5. Existence and uniqueness of stationary distributions
  6. Concerning the Feller property
  7. General remarks about fixed points of the mapping
  8. Fixed points existing by symmetry
  9. Symmetry in the limit
  10. Fixed points for $\pi_{1,2}$ and $\pi_{2,2}$
  11. Two elements -- the binomial case
  12. Finite spaces
  13. Non-uniform invariant distributions
  14. Uniform invariant distributions
  15. Finite sets in Euclidean space
  16. ...and 11 more sections

Key Result

Theorem 1

Consider a compact metric space $(S,d)$ and a probability distribution $\theta \in \mathcal{P}(S)$. For each pair $1 \le j \le k, \ k \ge 2$, the Markov chain $\mathbf{X}^{\theta,j,k} = (X^{\theta,j,k}(t), t = 0, 1, 2, \ldots)$ has a unique stationary distribution $\pi_{j,k}(\theta)$. From any initi and so there is convergence to stationarity in variation distance. Moreover, for $\pi = \pi_{j,k}(\

Figures (17)

  • Figure 1: Iterates of $\pi_{1,2}$ (left) and $\pi_{2,2}$ (right) on the unit interval from uniform initial distribution.
  • Figure 2: $S = \{a,b\}; \ k = 5, j = 4$. Iterates $n = 0,1,2,\ldots,10$. Left panel shows type (iii) behavior, Right panel shows the unstable fixed point at 0.17267.
  • Figure 3: $|S| = 4$, rank matrix $R$ at (\ref{['R2']}), $\pi_{1,2}$. Unstable behavior of iterates $\theta_i(n), n = 0,1,2,\ldots, 20$ in panels $i = 1,2,3,4$, starting from two different initial distributions $\theta^+, \theta^-$ near the fixed point.
  • Figure 4: A 9-point set in the plane. See text for explanation.
  • Figure 5: A BTL space $S$ with $|S| = 7$.
  • ...and 12 more figures

Theorems & Definitions (11)

  • Theorem 1: mappings_short Theorem 1
  • Proposition 2
  • proof
  • Lemma 3
  • Theorem 4: mappings_short Theorem 3
  • Conjecture 5
  • Lemma 6
  • Conjecture 7
  • Theorem 8: mappings_short Theorem 7
  • Theorem 9: mappings_short Theorem 4
  • ...and 1 more