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Partially exchangeable Markov chains and characterisation of multitype Lambda-coalescents

Adrián González Casanova, Noemi Kurt, Imanol Nuñez Morales, José Luis Pérez

TL;DR

This work introduces dice processes as the canonical model for consistent, partially exchangeable sequences of Markov chains on a finite state space and provides a complete characterization of admissible transitions via a decomposition into per-particle motion and coordinated updates. It develops a rigorous de Finetti-measure framework and a moment duality for the limiting frequency process, expressed as a stochastic differential equation driven by a Poisson random measure on the space of stochastic matrices. The results establish a precise link to multitype Lambda-coalescents with multiple switching by showing any such coalescent can be decomposed into independent coalescent dynamics plus a dice-type switching mechanism, with explicit integral representations for rate structures. By presenting a suite of concrete dice-process examples, the paper connects genealogical models in population genetics with interacting particle systems and stochastic exchange models, offering unified mathematical tools for convergence, duality, and coalescent analysis in multitype settings.

Abstract

In this paper, we study consistent and partially exchangeable sequences of Markov chains on a finite state space. We provide a characterisation of the admissible transition rates via a decomposition into individual and coordinated motion of particles. As a consequence, we find a characterisation of multitype Lambda-coalescents with multiple switches. Moreover, we provide convergence and duality results for the corresponding process of limiting relative frequencies that we call the de Finetti measure process, and discuss a number of examples from the recent literature.

Partially exchangeable Markov chains and characterisation of multitype Lambda-coalescents

TL;DR

This work introduces dice processes as the canonical model for consistent, partially exchangeable sequences of Markov chains on a finite state space and provides a complete characterization of admissible transitions via a decomposition into per-particle motion and coordinated updates. It develops a rigorous de Finetti-measure framework and a moment duality for the limiting frequency process, expressed as a stochastic differential equation driven by a Poisson random measure on the space of stochastic matrices. The results establish a precise link to multitype Lambda-coalescents with multiple switching by showing any such coalescent can be decomposed into independent coalescent dynamics plus a dice-type switching mechanism, with explicit integral representations for rate structures. By presenting a suite of concrete dice-process examples, the paper connects genealogical models in population genetics with interacting particle systems and stochastic exchange models, offering unified mathematical tools for convergence, duality, and coalescent analysis in multitype settings.

Abstract

In this paper, we study consistent and partially exchangeable sequences of Markov chains on a finite state space. We provide a characterisation of the admissible transition rates via a decomposition into individual and coordinated motion of particles. As a consequence, we find a characterisation of multitype Lambda-coalescents with multiple switches. Moreover, we provide convergence and duality results for the corresponding process of limiting relative frequencies that we call the de Finetti measure process, and discuss a number of examples from the recent literature.

Paper Structure

This paper contains 26 sections, 8 theorems, 151 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Exchangeable and partially exchangeable processes: Model and main results
  3. Notation, exchangeability and consistency
  4. Dice processes and the class of consistent and partially exchangeable Markov chains
  5. Consistency condition for the rates of dice processes
  6. De Finetti measure and moment duality
  7. Multitype L-coalescents: Characterisation in terms of dice processes
  8. Examples of dice processes
  9. Independent Markov chains
  10. Totally dependent Markov Chains
  11. Discrete exchange models
  12. Multinomial splitting process
  13. Dirichlet splitting process
  14. Harmonic splitting process
  15. Discrete multiagent instant exchange model
  16. ...and 11 more sections

Key Result

Lemma 2.1

For every $A$ and $\nu$ satisfying the conditions of Definition def:dice_process, there exists a $[d]^\infty$-valued process $X^{(\infty)}$ such that for each $n \in \mathbb{N}$, $X^{(n)}=R_n X^{(\infty)}$ is an $n$-dice process with parameters $A$ and $\nu$.

Figures (1)

  • Figure 1: Example of a graph for an arbitrary particle. The dotted line correspond to times where coordination occurs. The red line corresponds to the path of a particle that started at position $2$.

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.1
  • proof
  • Definition 2.7
  • Theorem 2.1
  • ...and 13 more