A Gibbs sampler in a generalized sense, II

Abstract

We consider two new conditions for finite sequences of (finite) stochastic matrices. The Gibbs samplers in a generalized sense which satisfy these conditions have important properties, and thus became among the first our favorite chains - our interest is to design very fast Markov chains and having, if possible, other important properties. We show, in the finite case, that the probability distribution of a random vector with independent components is a wavy probability distribution with respect to the lexicographic order and n+1 partitions which will be specified, where n is the dimension of random vector. We define the wavy probability distributions in a generalized sense. When these probability distributions have normalization constant, we give, under certain conditions, a formula to compute this constant. To give other examples of wavy probability distributions and of wavy probability distributions in a generalized sense, we consider the Potts model (a model used in statistical physics and other fields). Moreover, the normalization constant for the Ising model on C_{n}, the cycle graph with n vertices, is computed.