Asymptotic Properties For The Parameter Estimator In The AR(1) Process With MA(1) Noise

Probability is an important subject and focuses on studying regularity of random phenomenon, which is extensively used in natural science, technological science, social science, managerial science and so on. Probability limit theory is one of the branches and an important theoretical basis of Probability and Statistics. The large deviation theory mainly studies the problem of the ergodic convergence speed. Moreover, it has been one of the major branches in probability theory and has many applications in partial differential equations, Markov processes, statistics, insurance and finance. The future value can be predicted from the past value and present value in the time series. In the economic forecasting, it considers the dependence on the economic phenomena in time series and the interference of random fluctuations. So the accuracy in prediction of tendency is very high, such as the prediction of financial revenue and personal insurance growth.This article includes the following four chapters.In the first chapter, at first, we briefly review the basic concepts and results in Central Limit Theorem for Martingales and large deviation theory. Then we introduce some known conclusions about several other models related to our model. Finally, we give the motivations of this thesis.In the second chapter, we introduce the model:the AR(1) process with MA(1) noise, and the results obtained in this thesis are stated.In the third chapter, it shows the asymptotic distributions of?n under the given four different situations. Then the Central Limit Theorem for Martingales and splitting techniques are applied to conclude the asymptotic distributions of ?nIn the fourth chapter, we consider the moderate deviations for the parameter estimator only for the near-stationary case. At first, by giving the truncation of the martingale and the moderate deviations, we can get the moderate deviations for the parameter estimator.