Ergodicity Of The Time Series Model Class Of Doubly Stochastic Environment

Time series analysis is a branch of applied probability and statistics. Now it has become an indispensable tool for data analysis in every field of natural science and social science. Because of its comprehensive application in many fields, the study of the time series models is becoming more and more important and many researchers have obtained some excellent achievements. Among these studies, it is one of the most important components that studied the probability properties of these models themselves and the limitation of iterative series constructed by such time series models.According to the documents about the time series model in the past, whether it was in the linear time series model or not, its disturbances had been supposed to be a single white noise series and there was a specially fixed delay constant in every model. There is an obvious limitation in it, in other words, it can't describe the fact that the system's disturbance or the delay in it change stochastically because of affection of various random factors. Professor Hou Zhengting of the Institute of Probability & Statistics (IPS), the Central South University, has brought forward the Random Environment Time Series Model (RETSM) firstly to try to resolve the problem that the disturbance would be affected by random factors, and have carried out a lot of researches and attained a series achievement. To resolve the problem that the delay would be affected by random factors, basing on Hou Zhenting's theory, my mentor put forward a time series model that its delay controlled by a Markov chain with finite states, as a generalization of the time series model.In this thesis, following the idea and method ahead, applying the Markovnization and the theories of general state space Markov Chain, I have studied some time series models with stochastic delay, and have deduced some sufficient conditions about the companion geometric ergodicity of these models.Four parts compose this thesis as following:In Chapter 1, introduces the research status in quo about the time series models.In Chapter 2, a basic knowledge of this thesis, briefly presents some basic notions of the general states Markov chain and the ergodicity of the Markov chain.In chapter 3, I first study the linear autoregressive model with a stochastic delay and get corresponding Markov chain theory, and give the definition of companion ergodicity and companion geometric ergodicity of this model. Second, I get a sufficient condition of the companion geometric ergodicity of this model by applying the theories constructed formerly.In Chapter 4 researches of the companion geometric ergodicity of a nonlinear time series model by applying the same method used in chapter 3.