Abstract: 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 = X X' /T, where X is an n×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 = n W_n (t) W_n (t)' dt , where W_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 φ(x) that decays as x^-3/2 when x → ∞. The limit corresponding to 𝒲 is a Feller-Pareto distribution. An illustrative application is provided.

Key words and phrases: Empirical spectral distribution, Feller-Pareto distribution, non-stationary time series, sample covariance, Stieltjes transform.