Efficient coupling on the circle

Abstract

We consider efficient coupling specifically treating reversible Markov chains on the circle. We also show that most Markov chains with an efficient coupling have an asymptotic monotone function. Introduction This paper is prompted by a recent paper of [BK] concerning efficient couplings of irreducible reversible continuous time Markov chains on a finite state space 5. (We note those authors also studied efficiency questions for reflecting Brownian motion.) The starting point for this paper is the eigenvector expansion for the density function Pt(x,y) = Px(Xt = y) where (Xt)t>o is our reversible Markov chain ft(s.y) = p(?)Se~???F?(?)F?(?) ? t Here p is the (unique) invariant distribution and f? are the right eigenvectors satisfying Q i = ? Xi i i = 1 ? ? ? ? Sf?(?)p(?)F?(?) = ?ij, where Q is the Q-matrix or generator of X. We refer to [AF] for important background materials on reversible Markov chains. Of course if the f? are ordered according to increasing A?, then ?? = 0, f = 1, and we have that Slft(*,y)-ir(y)|=0(e-A'*) and ?) \pt(x,y) n(y) = o(e Xit) if and only if f?(?) = 0 for each eigenvector corresponding to A2.