The Limiting Spectral Density Of The Sample Covariance Matrix From Causal AR(1) Model

In the past decades, a significant advancement in the world has been in the rapid developments and wide applications of computer techniques. The rapid developments in computer techniques bring about possibility in collecting, storing and analyzing data of huge amount and large dimension. Classical limiting theorems perform very poorly or even are inapplicable for many large dimensional problems, especially when the dimension increases proportionally with sample size. Base on Theorem 1.1 of Bai and Zhou's "Large Sample Covariance With column Independence structure" , limiting spectral density(LSD) of large sample covariance matrices under dependence conditions is derived. As applications, suppose that X is a sample of size n from a causal AR(1) model , the Stieltjes transform of the LSD of sample covariance matrices is a root of quartic equation. We will solve the equation and plot the picture of LSD.