Optimal coadapted coupling for a random walk on the hyper-complete graph

Abstract

The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks on Z₂^d was considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such coadapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal coadapted coupling for the continuous-time symmetric random walk on K_n^d, where K_n is the complete graph with n vertices. Moreover, we show that although this coupling is not maximal for any n (i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling as n → ∞.