The Extension Of The Generalized Entropy Ergodic Theorem For Nonhomogeneous Markov Chains

Markov process is one of the most important probabilistic processes in the study of probability theory,and it is widely used in the social sciences such as computational science,stochastic fractal,economics,medicine,industry and so on.In the recent years,Wang and Yang have given the generalized entropy for nonhomogeneous Markov chains and studied the generalized entropy ergodic theorem for nonhomogeneous Markov chains.In this paper,we will promote Yang’s results to a class of function of two variables and prove the generalized entropy ergodic theorem for second order nonhomogeneous Markov information source by the method of martingale.Firstly,we briefly introduce some background of Markov process and the main research achievements both at home and abroad,we also recommend the structure of the paper.Then,this paper introduces some basic knowledge and makes a list of some important lemmas which are used in the study of Markov chains.Moreover,this paper recommends the generalized entropy for nonhomogeneous Markov chains and Yang’s the generalized entropy for nonhomogeneous Markov chains.Then we generalize the results of the generalized entropy ergodic theorem for nonhomogeneous Markov chains to a class of function of two variables.What’s more,we often describe the actual problems via second order nonhomogeneous Markov information source in the living.Yang and Liu have given some limit theorems for second order nonhomogeneous Markov information source.So we give the generalized entropy ergodic theorem for second order nonhomogeneous Markov information source by the use of Yang’s method.Finally,we summarize all above the paper,elaborate some deficiencies of the paper,and show the future directions for the research.