On singular value distribution of large-dimensional autocovariance matrices

Abstract

Let (εj)j≥0(εj)j≥0 be a sequence of independent pp-dimensional random vectors and τ≥1τ≥1 a given integer. From a sample ε1,…,εT+τε1,…,εT+τ of the sequence, the so-called lag-ττ auto-covariance matrix is View the MathML sourceCτ=T−1∑j=1Tετ+jεjt. When the dimension pp is large compared to the sample size TT, this paper establishes the limit of the singular value distribution of CτCτ assuming that pp and TT grow to infinity proportionally and the sequence has uniformly bounded (4+δ)(4+δ)th order moments. Compared to existing asymptotic results on sample covariance matrices developed in random matrix theory, the case of an auto-covariance matrix is much more involved due to the fact that the summands are dependent and the matrix CτCτ is not symmetric. Several new techniques are introduced for the derivation of the main theorem.