Spectral distribution of the sample covariance of high-dimensional time series with unit roots

Abstract

The aim is to study the empirical spectral distributions of two sample-covariance-type matrices associated with high-dimensional time series with unit roots. The first matrix is $S = XX'/T$; where $X$ is an $n imes T$ data matrix with rows represented by $n$ i.i.d. copies of $T$ consecutive observations of a difference-stationary process. The second matrix is $W = nint_0^1 B_n (t)B_n (t)' dt$; where $B_n(t)$ is an $n$-dimensional vector with i.i.d. Brownian motion components. We show that, as $n$ and $T$ diverge to infinity proportionally, the two distributions weakly converge to non-random limits. The limit corresponding to $S$ has a density $f(x)$ that decays as $x^{-3/2}$ when $x oinfty$. The limit corresponding to $W$ is a Feller-Pareto distribution.