Moment approach for singular values distribution of a large auto-covariance matrix

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) auto-covariance matrix of ε t . Since X T is not symmetric, we consider its singular values, which are the square roots of the eigenvalues of X T X T T . Therefore, the purpose of this paper is to investigate the limiting behaviors of the eigenvalues of X T X T T in two aspects. First, we show that the empirical spectral distribution of its eigenvalues converges to a nonrandom limit F . Second, we establish the convergence of its largest eigenvalue to the right edge of F . Both results are derived using moment methods.

Identifying the number of factors from singular values of a large sample auto-covariance matrix

“…Other related references include Jin et al [15] and Wang et al [26] where the limiting spectral distribution and the strong convergence of extreme eigenvalues are derived for the matrix 1 2 ( Σ ε + Σ ε ). One should mention that these works are more related to the study in Li et al [1] and Wang and Yao [27] on spectral limits of the matrix M ε , and they have no results either on convergence of the spiked (factor) eigenvalues or on the estimation of the number of factors as proposed in this paper.…”

“…For the strong consistency of the proposed ratio estimator k, a main ingredient is the almost sure convergence of the largest eigenvalue of the base matrix M ε to the right edge b, recently established in Wang and Yao [27]. This result serves as the cornerstone for distinguishing between significant factors and noise components.…”

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

“…This means that all the edges in the graph M (A(k, k + 1)) is repeated exactly twice, so the part of expectation Second, the number of isomorphism class in M (A(t, s)) (with each edge repeated at least twice in the original graph Q(i, j)) is given by the notation f t−1 (k) in Wang and Yao (2014), where…”

“…2014/10/16 file: autocross.tex date: October 14, 2018 eigenvalues. For the singular value distribution of X T , the limit (LSD) has been established in Li et al (2013) using the method of Stieltjes transform and in Wang and Yao (2014) using the moment method. The latter paper also establishes the almost sure convergence of the largest singular value of X T to the right edge of the LSD, thanks to the moment method.…”

“…In this paper, we investigate the same questions as in Wang and Yao (2014) but under a different asymptotic regime, the so-called ultra-dimensional regime where p → ∞, T → ∞ and p/T → 0.…”