A Strong Limit Theorem Of Harmonic Mean Of Transition Probability For Non-homogeneous Bifurcating Markov Chains Indexed By A Binary Tree

The research on the stochastic process indexed by trees aims to popularize and apply the classic theory in stochastic process indexed by a tree graph with a certain structure.With the deepening of research in recent years,the structural properties and limit properties of many complex systems such as tree graphs or tree networks have become hot topics in research,especially in the field of tree-index Markov chains,A large number of domestic and foreign scholars have obtained numerous significant research results,including the law of large numbers,central limit theorem,entropy ergodic theorem and many other limit theorems.The non-homogeneous bifurcating Markov chain indexed by a binary tree is a special kind of tree-index Markov chain,and its limit properties have been extensively studied by scholars and applied to many fields such as biodynamics and information theory.In recent years,the strong limit theorem about the harmonic mean of transition probability has been widely discussed.This thesis is devoted to the study of the limit properties of the transition probability for non-homogeneous bifurcating Markov chains indexed by a binary tree.The main contents are as follows:First of all,this thesis points out that for non-homogeneous Markov chains taking values in a finite state space,the strong limit theorem of the harmonic mean of random transition probability is a corollary of the strong limit theorem of any random adaptation sequence,and the research ideas are applied to give a new proof about the strong limit theorem of harmonic mean of the random transition probability for tree-index non-homogeneous Markov chains.This result is a generalization of the limit properties of random transition probability for non-homogeneous Markov chains.Secondly,in this thesis,we mainly study the strong limit theorem of the harmonic mean of transition probability for non-homogeneous bifurcating Markov chain indexed by a binary tree taking value in a finite state space.And the above result is generalized to the M-bifurcating Markov chain indexed by a M-branch Cayley tree.Meanwhile,by using the equivalence between the above models and the non-homogeneous tree-index Markov chain,the connection about the strong limit theorems of harmonic mean of their transition probabilities is discussed.Finally,this thesis also gives some applications for the above model's strong limit theorems of harmonic mean of their transition probabilities to illustrate their theoretical value.The content mainly involves using the above strong limit theorems of harmonic mean of their transition probabilities to derive the Borel's strong law of large numbers with the parameter p and the problem of estimating the upper limit of the arithmetic mean of the random transition probability for those models.