LLN for Quadratic Forms of Long Memory Time Series and Its Applications in Random Matrix Theory

We obtain a weak law of large numbers for quadratic forms of a stationary regular time series, imposing no rate of convergence to zero of its covariance function. We show how this law can be applied in proving universality properties of limiting spectral distributions of sample covariance matrices. In particular, we give another derivation of a recent result of Merlevède and Peligrad, who studied sample covariance matrices generated by IID samples of long memory time series.