Limit Theorems Based On Random Environments And Complex Networks

In this paper, we maily study several limit theorems based on random environments and complex networks. First, we prove the escape speed of persistent random walk in a random environment satisfies the principle of large deviation by applying the methods of splitting of hitting time and measure transform. Then, we introduce the model of multi-dimensional continuous-time random walk in a random environment with holding times. In order to show the asymptotic behaviors such as the law of large number and central limit theorem of escape speed, we construct the renewal structure under general Kalikow'condition, which came from the point "environment viewed from the particle". By virtue of the shape theorem of first-passage percolation and related models, we obtain the explicit expression of escape speed. What's more, we explain the relation between escape speed and Lyapounov exponents of random walk. Further, we extend the traditional web tree-graph model with preferential attachment in virtue of the recurrence equation of degree distribution. For the purpose of show the law of large number and central limit theorem on degree distribution, we construct a martingale. Furthermore, we prove degree distribution satisfies the principle of moderate deviation by computing general Cramer function of degree distribution and combining the properties of martingale. The law of iterated logarithm is also obtained. Finally, we discuss the principle of moderate deviation on ball distribution of Pólya urn model in a random environment by constructing martingale and applying numerical value approximation.