Segregating Markov chains (Q1800944)

If the distribution of a Markov chain converges to some distribution \(\pi \) irrespectively of initial distribution, the speed of convergence is measured by means of the variation distance. Sometimes it is more convenient to consider \(\sup \| P^n(x,\cdot)-P^n(y,\cdot)\| _{TV}\). \textit{O. Häggström} [Random Struct. Algorithms 18, No. 3, 267--278 (2001; Zbl 1002.60096)] gave an example of a finite reducible Markov chain with the property: two copies of the chain, starting in different states \(x\) and \(y\), can be coupled in a way that they meet in a finite time but the total variation distance never goes down a fixed positive value, this is called the segregation of two states. The paper continues the investigation of this phenomenon. Namely, it considers two coupled copies of a Markov chain on a countable state space starting from the points \(x\) and \(y\), they meet after time \(\tau \). If the two copies can be coupled so that \(P\{\tau <\infty \}=1\), then \[ \sup \lim_{n\to \infty }\| P^n(x,\cdot)-P^n(y,\cdot)\| _{TV}\leq 1/2, \] where the supremum is taken over transition matrices \(P\) and states \(x\) and \(y\).