On singular value distribution of large-dimensional autocovariance matrices.

Let ( ε j ) j ≥ 0 be a sequence of independent p -dimensional random vectors and τ ≥ 1 a given integer. From a sample ε 1 , … , ε T + τ of the sequence, the so-called lag- τ auto-covariance matrix is C τ = T − 1 ∑ j = 1 T ε τ + j ε j t . When the dimension p is large compared to the sample size T , this paper establishes the limit of the singular value distribution of C τ assuming that p and T grow to infinity proportionally and the sequence has uniformly bounded ( 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 C τ is not symmetric. Several new techniques are introduced for the derivation of the main theorem. [ABSTRACT FROM AUTHOR]