Mixing rates for potentials of non-summable variations

Mixing rates, relaxation rates, and decay of correlations for dynamics defined by potentials with summable variations are well understood, but little is known for non-summable variations. This paper exhibits upper bounds for these quantities for dynamics defined by potentials with square-summable variations. We obtain these bounds as corollaries of a new block coupling inequality between pairs of dynamics starting with different histories. As applications of our results, we prove a new weak invariance principle and a Hoeffding-type inequality.

Diagonal couplings of quantum Markov chains

In this paper we extend the coupling method from classical probability theory to quantum Markov chains on atomic von Neumann algebras. In particular, we establish a coupling inequality, which allow us to estimate convergence rates by analyzing couplings. For a given tensor dilation we construct a self-coupling of a Markov operator. It turns out that the coupling is a dual version of the extended dual transition operator studied by Gohm et al. We deduce that this coupling is successful if and only if the dilation is asymptotically complete.

Optimal Coadapted Coupling for a Random Walk on the Hyper-Complete Graph

The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks on Z2d 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 Knd, where Kn 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 → ∞.

Optimal Coadapted Coupling for a Random Walk on the Hyper-Complete Graph

The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks onZ2dwas 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 onKnd, whereKnis the complete graph withnvertices. Moreover, we show that although this coupling is not maximal for anyn(i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling asn→ ∞.

Rates of Poisson convergence for some coverage and urn problems using coupling

Bounds on the rate of convergence measured by the variation distance are obtained for the number of large spacings and for two occupancy problems connected with multinomial and Pólya sampling. The bounds are derived by imbedding techniques together with the elementary coupling inequality.

Rates of Poisson convergence for some coverage and urn problems using coupling

Bounds on the rate of convergence measured by the variation distance are obtained for the number of large spacings and for two occupancy problems connected with multinomial and Pólya sampling. The bounds are derived by imbedding techniques together with the elementary coupling inequality.