Asymptotic Distributions And Moderate Deviations For The Parameter Estimators Related To The Near-stationary Second Order Autoregressive Process

Probability theory focuses on studying the regularity of random phenomenon, which is extensively used in natural science, social science, practical production and so on. The large deviation theory beginning at 1960's, studies the problem of the ergodic convergence speed. Nowadays, 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. In finance and economic, the change of stock price roughly present autoregressive process. By the analysis of the parameter estimator, we can study the law of the data sequence to effectively control and forecast the stock price.This article includes the following four chapters.In the first chapter, at first, we briefly review the basic concepts and results in large deviation theory. Then some known conclusions are stated. Finally, we give the motivations of this thesis.In the second part, the model studied in the paper is introduced, and the results obtained in this thesis are stated.In the third chapter, we give the proof of the asymptotic distributions for the parameter estimators ?n, ?n and the Durbin-Watson statistic Dn. Firstly, we find the equivalent fractional form of ?n-?n* so that we can make use of martingale central limit theory and weak law of large numbers to obtain the asymptotic distributions of nominator and denominator respectively. Then, by Slutsky's lemma, the asymptotic distribution of ?n can be easily obtained. Secondly, the central limit theory of two-dimensional martingale and splitting techniques are applied to conclude the joint asymptotic distribution of ??n, ?n?. Finally, the asymptotic distribution of Dn can be concluded using the conclusion of ?nIn the fourth chapter, we prove the moderate deviations for ?n, ?n and the Durbin-Watson statistic Dn. At first, by exponential equivalence, the moderate deviations for triangular array of martingale differences and splitting techniques, we can formulate the moderate deviations for ?n and ?n. At last, we have no difficulty obtaining the moderate deviations of Dn by the conclusion of ?n.