The convergence rate of the Gibbs sampler for generalized $1-$D Ising model

The rate of convergence of the Gibbs sampler for the generalized one-dimensional Ising model is determined by the second largest eigenvalue of its transition matrix in absolute value denoted by $\beta^*$. In this paper we generalize a result from Shiu and Chen $(2015)$ for the one-dimensional Ising model with two states which gives a bound for $\beta^*$ . The method is based on Diaconis and Stroock bound for reversible Markov processes. The new bound presented in this paper improves Ingrassia's $(1994)$ result.