The Research On Ergodic Theorem For Non-homogeneous Markov Chains And MCMC Algorithms

With the rapid development of computer technology, people rely on computers to do a lot of random experiments, while the feasibility of sampling from the complicated posterior probability distribution is derived from the good properties of Markov chains. As a result, the MCMC algorithm is a powerful tool to deal with the practical problems that the traditional methods are often hard to deal with.MCMC algorithms mainly include Metropolis-Hastings algorithm and Gibbs algorithm. Metropolis-Hastings algorithm is an improvement of the traditional MCMC algorithm,and the Gibbs algorithm is the amendment of Metropolis-Hastings algorithm, and it is more suitable for the situation of high dimension. Metropolis-Hastings algorithm and Gibbs algorithm have its improved form as well as convergence analysis and identification.We can construct the MCMC algorithm according to the practical problems, such as the adaptive MCMC algorithm, etc.. However, wether these Markov chains we constructed is converge to the stationary distribution and the convergence speed need to be studied in depth. Geometric ergodic is one of the important properties of Markov chains. The Markov chains which satisfy the drift condition and minorisation condition are geometric ergodic,which has an essential effect on the quantitative convergence rate.In this paper, we mainly discuss the MCMC algorithm and the related properties and conclusions of homogeneous Markov chains, including the convergence rate and the central limit theorem of the Markov chain. The homogeneous theories is extended to the non-homogeneous Markov chain, and the ergodic theorem and the related inference of the non-homogeneous Markov chain are given under certain measurements. In addition, in problem of non-homogeneous Markov chain quantitative convergence rate, we use the total variation norm and the coupling and drift conditions to give the estimation of upper bound of the convergence. Finally, the paper gives two examples to verify effectiveness of the theorem.