A Representation for Transition Probability of High-order Markov Chain

In many applications,the high-order Markov model is an important tool to study the long-term correlation.In order to overcome the shortcomings that the existing model for high-order Markov can not engage dynamics,the representation of transition probability from initial state to any state is discussed,it facilitates the study of dynamic data.And the expression of stationary distribution is given too,which provide a tool for steady-state study.Since the result is similar to first order Markov chain,the difficulty is greatly reduced while applying higher-order Markov chain to various fields.