# Non-coupling from the past

## TL;DR

This paper investigates the conditions under which trajectories of finite state space Markov chains coalesce, introducing the coalescence number and analyzing the set of such numbers for given transition matrices.

## Contribution

It introduces the coalescence number of a Markovian coupling and analyzes the set of coalescence numbers for a given transition matrix, advancing understanding of coupling behavior.

## Key findings

- Defined the coalescence number $k(rac)$ for Markovian couplings.
- Analyzed the set $K(P)$ of coalescence numbers for a transition matrix $P$.
- Provided results on the conditions for coalescence and non-coalescence of trajectories.

## 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' $k(\mu)$ of a Markovian coupling $\mu$ 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.