The Limit Properties Of Random Walk In Time-Random Environment On The Line And Branching Processes With Immigration In Random Environment

Random walk in random environments that developed in recent years has become an active branch of stochastic processes. It has deep realistic background and potential application. A lot of problems in nature science are related with random walk. Since Solomon introduce the concept of random walk in one-dimensional independent random environments, it has been the hot spot of theoretical researching. Kalicow produce a general model of random walk in d-dimensional independent random environments. There are a lot of limit theorems about random walk in these models over recent years. When it has been developing, Cogburn and Orey introduce another concept of Markov chain in random environments, which discusses the time- random environments. Such concept is different from space-random environments in Solomon's model.We study a specific model of random walks in time-random environments in this thesis. If the environments are stationary and ergodic, we derive a law of large numbers and a center limit theorem of this random walk under some reasonable conditions. Especially, when the environments are independent identically distributed, there are more exact results which are similar to the corresponding results in the case of the classical law of large numbers and the center limit theorem. Branching processes in random environments also is a hotspot in the research of stochastic processes, and branching processes with immigration in random environments is one of the current great potential research trends about it. We study some properties of time sequence estimator including non-asymptotic property, uniformly normality and basic limit property in branching processes with immigration in random environments at this thesis.