Strong Limit Theorems For Stochastic Process Indexed By A Tree

The stochastic process indexed by a tree is one of the research directions in probability theory, and it has been widely applied to many disciplines. In the field of biology, for example, biologists summarized the rod-shaped bacteria’s division rules:one bacterium is devided into two equal parts, if every rod-shaped bacterium’s lifetime is considered as a random variable, the whole of them can be considered as a tree-indexed stochastic process, so the research of tree-indexed stochastic process is an important topic in the filed of stochastic process, and its limit theorems have very important theoretical and practical significance.In recent years, Professor Yang Weiguo has studied the strong laws of large numbers and Shannon-McMillan theorems for the tee-indexed Markov chains, a great deal of papers have been published in the national journals and international journals. This doctoral dissertation is on the basis of Yang’s research methods, further study the strong laws of large numbers and Shannon-McMillan theorems of the tee-indexed Markov chains taking values on countable space.Hidden Markov tree models were introduced by Crouse et al., but their definition is not strict. In this doctoral dissertation, we will give the rigorous definition for hidden Markov tree model, and obtain some properties for hidden Markov tree model, as a corollary, we will point out the existence of hidden Markov tree models. Finally we will obtain the strong laws of large numbers for hidden Markov tree models indexed by a Cayley tree, as corollaries, we will obtain some strong laws of large numbers for the parameters and the Shannon-McMillan theorem for this model.There are five chapters in this doctoral dissertation.In chaper1, we first give an introduction for the basic notations and review the theoretical results have been obtained about the tree-indexed Markov chains, then we will introduce the different definitions of hidden Markov chain in the different disciplines and prove their equivalence.In chapter2, the first secction, we study the strong laws of large numbers for the frequencies of occurrence of states and ordered couples of states for countable Markov chains indexed by a Cayley tree, which extend and improve the corresponding results in the finite states case. The second section, using the similar approach, we study the strong laws of large numbers for the frequencies of occurrence of states and ordered couples of states for countable Markov chains indexed by a uniformly bounded tree.In chapter3, we establish a strong law of large numbers for the bivariate functions of countable Markov chains indexed by a Cayley tree. As corollaries we get the strong laws of large numbers for frequencies of occurrence of states and ordered couples of states for countable Markov chain indexed by a Cayley tree, and get the Shannon-McMillan theorem for In chapter4, we study the hidden Markov tree model, give the vigorous definition for this model, and obtain some properties for them, as a corollary, we will prove the existence of hidden Markov tree models.In chapter5, we study the strong laws of large numbers for hidden Markov tree models indexed by a Cayley tree, and as collaries, we obtain some strong laws of large numbers for the parameters and the Shannon-McMillan theorem for this model.