Limit Theorems Based On Random Elds And Complex Networks

In this thesis, we mainly study the limit theorems based on random fields indexed by trees and complex networks. There are six chapters in this doctoral dissertation.In chapter 1, we will present the basic theory and the background of random fields indexed by trees and complex networks. The main results obtained and the main methods we used in this thesis are stated.In chapter 2, we give the definition of centered random field indexed by trees and study the central limit theorem of such case.In chapter 3, we give the definition of quasi-associated random fields indexed by trees and study the central limit theorem of it.In chapter 4, at first we study the strong law of large numbers and Shannon theo-rem on homogeneous Markov chain fields indexed by uniformly bounded trees, then we obtain the similar results of non-homogeneous Markov chain fields indexed by uniformly bounded trees by using the similar method as homogeneous case. By using the central limit theorem of martingale sequence, we also get the central limit theorem of stationary Markov chain fields indexed by uniformly bounded trees. At last, we give the weak large deviation principle of empirical pair measures of homogeneous Markov chain fields indexed by uniformly bounded trees.In chapter 5, we put forward a model of non-homogeneous growing network, then we consider the limit properties in connection with the topology of the network such as the degree distributions and the degree correlations, when the number of nodes of our network tends to infinity.In chapter 6, we use utility to describe the attractive effect and then study a simple asymmetrical evolving model, considering both preferential attachment and the random-ness of the utility.