The Perturbation Bounds of the Joint Stationary Distribution Vector of the High-order Multivariate Markov Chains

The perturbation bounds of the joint stationary distribution vector of the high-order multivariate Markov chains are established. By the properties of the left and right eigenvectors of the probability transition matrix of the high-order multivariate Markov chains, the perturbation bound of the joint stationary distribution vector of the high-order multivariate Markov chains is obtained, which generalizes the results of the existing perturbation bounds of the joint stationary distribution vector of one-order multivariate Markov chains. Then the computational perturbation bound is given by the characteristic of the probability transition matrix of the high-order multivariate Markov chains, which also generalizes the corresponding perturbation bound of the joint stationary distribution vector of one-order multivariate Markov chains. Moreover, considering the perturbation in the item of the joint stationary distribution vector of high-order multivariate Markov chains, the perturbation bound of the joint stationary distribution vector based on component form is established by Paz's inequality, to observe the perturbation of a state in a chain of the high-order multivariate Markov chains.