Spectral Distribution of the Sample Covariance of High-Dimensional Time Series With Unit Roots

Saturday 1st January 2022
Statistica Sinica, forthcoming
Onatski, A. and Wang, C.
We 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 x T data set, with rows represented by n independent and identically distributed (i.i.d.) copies of T consecutive observations of a difference-stationary process. The second matrix is W = [code], where Wn (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 [code] that decays as [code] when [code]. The limit corresponding to W is a Feller-Pareto distribution. An illustrative application is provided.
Empirical spectral distribution
sample covariance
non-stationary time series
Stieltjes transform
Feller-Pareto distribution