Non-coupling from the past

TL;DR

Non-coupling from the past analyzes when coupling from the past (CFTP) yields coalescence in finite-state Markov chains by introducing the coalescence number $k(\mu)$ for a Markovian coupling and the set $K(P)$ of achievable coalescence numbers. The authors connect coalescence properties to representations of $P$ via convex combinations of extremal mappings, block-lumpability through $\mathcal{S}$-block measures, and a matrix-rank characterization of $k(\mu)$. They establish foundational results: $1\in K(P)$ for aperiodic $P$, $n\in K(P)$ for doubly stochastic $P$, and detailed behavior for equal-entry chains $P_n$, including divisors of $n$ appearing in $K(P_n)$ and the exclusion of $n-1$. Across examples and general constructions, the work clarifies how coupling choices influence exact sampling via CFTP and guides the design of couplings with targeted coalescence properties.

Abstract

The method of 'coupling from the past' permits exact sampling from the invariant distribution of a Markov chain on a finite state space. The coupling is successful whenever the stochastic dynamics are such that there is coalescence of all trajectories. The issue of the coalescence or non-coalescence of trajectories of a finite state space Markov chain is investigated in this note. The notion of the 'coalescence number' $k(μ)$ of a Markovian coupling $μ$ is introduced, and results are presented concerning the set $K(P)$ of coalescence numbers of couplings corresponding to a given transition matrix $P$. Note: This is a revision of the original published version, in which part of Theorem 6 has been removed. A correction may be found in Thm 5.3 of arXiv:2510.13572.

Figures (2)

• Figure 1: An illustration of coalescence of trajectories in CFTP with $|S|=5$.
• Figure 2: Diagrammatic representations of the four functions $f_i$ of Example \ref{['ex:7']}. The corresponding equivalence classes are not ${\bm{\mu}}$-a.s. constant.

Theorems & Definitions (28)

• definition 1
• theorem 1: PW2
• theorem 2
• proof
• lemma 1
• proof
• lemma 2
• proof
• definition 2
• theorem 3