Decomposition Theorems Of Design Matrices For General Autoregressive Models With Applications To Time Series Analysis

In this paper, we firstly introduce general autoregressive models; secondly, mainly results are given. keystones are the section 3 and 4, which are prooves of main results and applications. Theorems arederived to decompose the design matrix(X(k), ..., X(k—p+1)), into a diagonal form with stationary, oscillatory and explosive components for general autoregressive submodels, that is, there exists a nonsingular (real) matrix R, such thatRPnR' = (1 + o(1))diag(Pn(3),Pn(2),Pn(1)), a.s.,where Pn(3),Pn(2) and Pn(1) correspond to the explosive, oscillatory and stationary submodels, respectively. Moreover,with d,r,R,M and m specified in lemmas and theorems. Futhermore, wo prove that under a complex transformation, Pn(3),Pn(2) and Pn(1) behave, a.s., approximately diagonally. Especially, one result obtained from prooves is more accurate than theorem 2 of Lai andWei[4].Next, based on these decomposition principles for the design matrices, we derive criteria to identify the types of autoregressive models.At last, these principles are used to give a complete answer considering convergence rate to the strong consistency of the Least-sqrares estimates θn of the known true parameter 9, that is, under a mild regularity condition,a rate result of the law of the iterated logarithm for general autoregressive models which violate the Lai and Wei's " weakest" condition,...