Some Strong Limit Theorems For Hidden Nonhomogeneous Markov Models

Markov process is an important kind of stochastic process.Since it was proposed by Soviet mathematician A.A.MapkoB in 1907,it has become a branch of mathematics through the effort of mathematicians all over the world.The concept of hidden Markov models is the generalization of the concept of classical Markov models.With the development and the application of hidden Markov models,people became more and more interested in the theories and applications of hidden Markov models.Hidden Markov models have been widely used for modeling sequences of weakly dependent random variables,as well as in speech processing,neurophysiology,biology,etc.Though people have made great progress in the study of hidden Markov models,the currently available theories couldn't resolve all the questions,because the actual problems are much more complex than the mathematical models,and the incidents observed and the models are not one-to-one.There is still much room for the improvement of its theoretical principles.Strong limit theorems plays a leading role in theoretical studies and has important application value.It's always a major problem in the study of probability theory.So the study of the strong limit theories for hidden Markov models has great significance.The purpose of this paper is to study the strong limit theorems for hidden nonhomogeneous Markov models by using the method of the martingale theory proposed by Yang Weiguo.In this paper,firstly,we introduce some knowledge on hidden Markov models,including the definition,the applications,the advantages,the equivalent definitions, some properties,relative entropy densities and strong Markov properties of hidden Markov models.Secondly,the strong limit theorem for countable hidden nonhomogeneous Markov models is obtained.As corollary,we generalize two known results.We also get some convergence theorems for countable hidden nonhomogeneous Markov models.Lastly,the strong limit theorems for the frequency of states' emergence,the strong laws of large numbers and limit theorem of relative entropy densities for finite hidden nonhomogeneous Markov models are obtained.