Strong Limit Theorems For Countable Nonhomogenous Mth-order Markov Chains And Application

Markov chain is a special random process with no after effectiveness, Strong limit theorems for Markov Chain are one basic field of Markov chain。The definition of higher order Markov chain is an extension of the definition of Markov chain.But,As the development and application of theorms for Markov chain.Many cases can't be solved or can't be solved well by first order Markov chain. But it will reach a very good effection by using higher order Markov chain theorms.Therefore, It is extremely necessary to the theoretical reseach and further application of higher order Markov chain.In this paper we principally make the following five aspects:(1) we prove two equivalent definition and two related properties of countable nonhomogenous mth-order Markov chains.(2) we establish the ergodicity problems of countable nonhomogenous mth-order Markov chains .we define the properties of strongly ergidic,absolute mean strongly ergodic,the convergence of cesaro average for countable nonhomogenous mth-order Markov chains,and we prove a sufficient condition of these properties(3) we obtain a class of strong limit theorems of functional average value and relative entropy density for countable nonhomogenous mth-order Markov chains.(4) we research the practical application of mth-order Markov chains.An improved mth-order Markov chain model and a mth-order multivariate Markov chain model will be constructed. To the estimate for parameter in the model,we put forward a kind of simple effective estimation method, we can construct these models by calculate state transition frequency matrix and solve some linear programming problems.(5) we explain how to construct mth-order markov chain model and mth-order multivariate mavkov chain model by a actual example of Categorical data sequences. Through the comparison of the prediction accuracy of these models,we can know that the mth-order multivariate Markov chain model is better than the first-order markov chain model or the mth-order Markov chain model. So that, the mth-order multivariate Markov chain model have strong research value and practical significance.