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Coalescence in Markov chains

Geoffrey R. Grimmett, Mark Holmes

TL;DR

This work analyzes coalescence phenomena for families of Markov chains {X^i} on a finite state space S under couplings that preserve the marginal transition structure P. It develops the grand-coupling framework, introduces the coalescence number k(μ) and the coalescence-set K(P), and connects forward coalescence to the coupling-from-the-past method. Central contributions include a detailed study of block measures arising from lumpability, criteria for their existence, and constructions of non-block measures showing that K(P) can contain nontrivial elements; plus an inverse problem characterizing which transition matrices can be generated from a given function family G. These results advance the theoretical understanding of coalescence, CFTP, and iterated random functions, with implications for exact simulation and the structure of coalescence classes.

Abstract

A Markov chain $X^i$ on a finite state space $S$ has transition matrix $P$ and initial state $i$. We may run the chains $(X^i: i\in S)$ in parallel, while insisting that any two such chains coalesce whenever they are simultaneously at the same state. There are $|S|$ trajectories which evolve separately, but not necessarily independently, prior to coalescence. What can be said about the number $k(μ)$ of coalescence classes of the process, and what is the set $K(P)$ of such numbers $k(μ)$, as the coupling $μ$ of the chains ranges over couplings that are consistent with $P$? We continue earlier work of the authors ('Non-coupling from the past', $\textit{In and Out of Equilibrium 3}$, Springer, 2021) on these two fundamental questions, which have special importance for the 'coupling from the past' algorithm. We concentrate partly on a family of couplings termed block measures, which may be viewed as couplings of lumpable chains with coalescing lumps. Constructions of such couplings are presented, and also of non-block measure with similar properties.

Coalescence in Markov chains

TL;DR

This work analyzes coalescence phenomena for families of Markov chains {X^i} on a finite state space S under couplings that preserve the marginal transition structure P. It develops the grand-coupling framework, introduces the coalescence number k(μ) and the coalescence-set K(P), and connects forward coalescence to the coupling-from-the-past method. Central contributions include a detailed study of block measures arising from lumpability, criteria for their existence, and constructions of non-block measures showing that K(P) can contain nontrivial elements; plus an inverse problem characterizing which transition matrices can be generated from a given function family G. These results advance the theoretical understanding of coalescence, CFTP, and iterated random functions, with implications for exact simulation and the structure of coalescence classes.

Abstract

A Markov chain on a finite state space has transition matrix and initial state . We may run the chains in parallel, while insisting that any two such chains coalesce whenever they are simultaneously at the same state. There are trajectories which evolve separately, but not necessarily independently, prior to coalescence. What can be said about the number of coalescence classes of the process, and what is the set of such numbers , as the coupling of the chains ranges over couplings that are consistent with ? We continue earlier work of the authors ('Non-coupling from the past', , Springer, 2021) on these two fundamental questions, which have special importance for the 'coupling from the past' algorithm. We concentrate partly on a family of couplings termed block measures, which may be viewed as couplings of lumpable chains with coalescing lumps. Constructions of such couplings are presented, and also of non-block measure with similar properties.
Paper Structure (7 sections, 17 theorems, 74 equations, 1 figure)

This paper contains 7 sections, 17 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Grand couplings of Markov chains
  3. Coalescence of trajectories
  4. Lumpability and block measures
  5. Existence of block measures
  6. Existence of non-block measures
  7. Functions that generate transition matrices

Key Result

Theorem 2.3

For $P\in\mathcal{P}_S$, we have $|\mathcal{L}_P|\ge 2$ if and only if $P$ has at least two rows each of which contains some entry lying in the open interval $(0,1)$.

Figures (1)

  • Figure 3.1: Diagrammatic representations of the four functions $f_{i,j}$ of Example \ref{['ex:7']}.

Theorems & Definitions (46)

  • Definition 2.1: ncftp
  • Example 2.2: Independence coupling
  • Theorem 2.3
  • proof
  • Remark 2.4: Random transition matrix
  • Theorem 2.5: Doeblin
  • Theorem 2.6: BirkhvN
  • Lemma 3.1
  • proof
  • Remark 3.2: Coupling from the past (CFTP)
  • ...and 36 more