Limiting Spectral Distribution of Large Sample Covariance Matrices Associated with a Class of Stationary Processes

In this paper, we derive an extension of the Marc̆enko–Pastur theorem to a large class of weak dependent sequences of real-valued random variables having only moment of order 2. Under a mild dependence condition that is easily verifiable in many situations, we derive that the limiting spectral distribution of the associated sample covariance matrix is characterized by an explicit equation for its Stieltjes transform, depending on the spectral density of the underlying process. Applications to linear processes, functions of linear processes, and ARCH models are given.