Limiting Spectral Distribution Of Linear Time Series

Sample covariance matrices play a very important role in multivariate statistical analysis and other fields. It is one of the earlier research directions and has lots of mature results in RMT. For a random matrix X, if its entries are independent of each other, then under certain conditions its sample covariance matrix has a limiting spectral distribution, which is subject to the MP law. When X’s entries are not independent, under the same conditions the sample covariance matrix of X also has limiting spectral distribution, which is not subject to the MP law. This article studies the limiting spectral distributions of linear time series under the condition of weakening independence, and obtains the corresponding density functions. This paper is organized as follows.In part one, we first introduce the background of large dimensional sample covariance matrix, research results and research methods. Then we introduce related contents of several linear time series.In part two, we research the limiting spectral distribution of short memory processes, including ARMA(p, q) process, AR(p) and MA(q) process. First of all, we prove short memory processes’sample covariance matrices have limiting spectral distributions. Secondly, we find out the corresponding density functions. Lastly, we solve the limiting spectral distribution of ARMA(1,1) process.In the last part, we study long memory process’s limiting spectral distribu-tion, mainly ARFIMA(p,q) process. We not only prove the existence of the limit spectral distribution and give the implicit expression, but also research correspond-ing density function at regional R\{0} and the support of limiting spectral distri-bution.