The Coupling Of The Markov Chain

Markov chain is widely applied especially in the recent twenty years. Benefited from the widely application of Monte Carlo method of the Markov Chain, the approximate calculation method of model, once was unfeasibly worked out because of the too much calculation, is feasibly now, hence more and more people want to get knowledge of the Markov Chain.Firstly some basic concept of Markov Chain was introduced briefly, and the Markov Chain existence theorem that plays a basic role in the Markov Chain theory was formulated using elementary method. The mixing time is an important unit in the Markov Chain Monte Carlo method. Secondly in this paper the coupling method of Markov Chain was introduced, and the coupling time was used to control the mixing time, and the nature of the mixing time was discussed using coupling method.Thirdly the error estimation theorem in the Markov Chain Monte Carlo method was formulated in detail using the mixing time and spectral gap and the coupling method in the paper.