The Strong Law Of Large Numbers For Second-order Countable Nonhomogeneous Hidden Markov Models

Hidden Markov model is a mathematical model based on probability and statistics theory,which can be used for statistical analysis with incomplete data.Hidden Markov model is used as an important research tool,which has wide range of application.Thus,research on hidden Markov has important practical significance.In practical applications,a simple first order hidden Markov model is used generally,but the accuracy of the model may be limited when solving the complex problem of state transfer.Therefore,this paper will make a further study of second-order countable nonhomogeneous hidden Markov models based on the first-order hidden Markov model.First of all,we introduce the background and significance of the present research state,the main research content of this article and the arrangement of the paper.Then,we introduce the preliminary knowledge related to the research of the subject,including the basic concepts,the definition and nature of conditional expectation,definition and properties of Markov chain,related properties and theorems,and so on.In the third chapter,according to the definition of the second-order discrete hidden Markov model,we get the definition of second-order countable nonhomogeneous hidden Markov models,we also get the equivalence property and its proof.In the fourth chapter,we get the strong law of large numbers for second-order countable nonhomogeneous hidden Markov models based on the strong law of large numbers for first-order hidden Markov model,and the detailed proof process is given.And two corollaries for application are shown at the end.The last one is conclusion which summarizes the whole article,and analyzes the scarcity and more researchful domain.