On singular values distribution of a matrix large auto-covariance in the ultra-dimensional regime

Abstract

Let (ε t ) t>0 be a sequence of independent real random vectors of p-dimension and let X T = s+T t=s+1 ε t ε T t−s /T be the lag-s (s is a fixed positive integer) autocovariance matrix of ε t . This paper investigates the limiting behavior of the singular values of X T under the so-called ultra-dimensional regime where p → ∞ and T → ∞ in a related way such that p/T → 0. First, we show that the singular value distribution of X T after a suitable normalization converges to a nonrandom limit G (quarter law) under the forth-moment condition. Second, we establish the convergence of its largest singular value to the right edge of G. Both results are derived using the moment method.

Spiked eigenvalues of high-dimensional sample autocovariance matrices: CLT and applications

“…So their proposed ratio-based estimator is also applicable in weak spiked cases when the spiked eigenvalues are of constant order. More result for unspiked eigenvalues are derived in Li et al [21], Wang and Yao [28] and Bose and Bhattacharjee [7]. Recently, Yao and Yuan [31],…”