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

Abstract

Let $({\epsilon }_t{)}_{t>0}$ be a sequence of independent real random vectors of $p$-dimension and let $X_T={\sum }_{t=s+1}^{s+T}{\epsilon }_t{\epsilon }_{t-s}^T/T$ be the lag-$s$ ($s$ is a fixed positive integer) auto-covariance matrix of ${\epsilon }_t$. 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.

29 pages, 8 figures

References in corpus (4)

• Factor modeling for high-dimensional time series: Inference for the number of factors
• On singular value distribution of large dimensional auto-covariance matrices
• On singular values distribution of a large auto-covariance matrix in the ultra-dimensional regime
• Strong limit of the extreme eigenvalues of a symmetrized auto-cross covariance matrix

Cited by in corpus (3)

• Identifying the number of factors from singular values of a large sample auto-covariance matrix
• On eigenvalue distributions of large auto-covariance matrices
• Order Determination for Spiked Models