[Paper Review] Spectral analysis of linear time series in moderately high dimensions

Lili Wang, Alexander Aue+1 more|arXiv (Cornell University)|Apr 23, 2015

Random Matrices and Applications8 references3 citations

TL;DR

This paper establishes the almost sure convergence of the empirical spectral distribution of renormalized and symmetrized sample autocovariance matrices in moderately high-dimensional linear time series, where dimension $ p $ and sample size $ n $ both tend to infinity with $ p/n \to 0 $. Under assumptions of i.i.d. innovations with finite fourth moments and Hermitian, simultaneously diagonalizable coefficient matrices, the limiting spectral distribution is characterized via the Stieltjes transform and corresponds to a nonrandom limit on the real line.

ABSTRACT

This article is concerned with the spectral behavior of $p$-dimensional linear processes in the moderately high-dimensional case when both dimensionality $p$ and sample size $n$ tend to infinity so that $p/n o0$. It is shown that, under an appropriate set of assumptions, the empirical spectral distributions of the renormalized and symmetrized sample autocovariance matrices converge almost surely to a nonrandom limit distribution supported on the real line. The key assumption is that the linear process is driven by a sequence of $p$-dimensional real or complex random vectors with i.i.d. entries possessing zero mean, unit variance and finite fourth moments, and that the $p imes p$ linear process coefficient matrices are Hermitian and simultaneously diagonalizable. Several relaxations of these assumptions are discussed. The results put forth in this paper can help facilitate inference on model parameters, model diagnostics and prediction of future values of the linear process.

Motivation & Objective

• To analyze the spectral behavior of $ p $-dimensional linear processes when both dimension $ p $ and sample size $ n $ are large, with $ p/n \to 0 $.
• To establish the almost sure convergence of the empirical spectral distribution (ESD) of renormalized and symmetrized sample autocovariance matrices to a nonrandom limit distribution.
• To characterize the limiting spectral distribution (LSD) of these matrices under structural assumptions on the coefficient matrices and innovation distribution.
• To provide a foundation for inference on model parameters, diagnostics, and prediction in high-dimensional time series.
• To extend classical results like the semi-circle law (Bai & Yin, 1988) to the case of dependent, high-dimensional time series with temporal and cross-sectional dependence.

Proposed method

• Define the symmetrized lag-$ \tau $ sample autocovariance matrix $ \mathbf{S}_{\tau} $ and its population counterpart $ \mathbf{\Sigma}_{\tau} $, then form the renormalized matrix $ \mathbf{C}_{\tau} = \sqrt{n/p}(\mathbf{S}_{\tau} - \mathbf{\Sigma}_{\tau}) $.
• Assume the linear process is driven by i.i.d. $ p $-dimensional innovations with zero mean, unit variance, and finite fourth moments.
• Impose structural assumptions: coefficient matrices are Hermitian and simultaneously diagonalizable, ensuring joint spectral structure.
• Use the Stieltjes transform to characterize the limiting spectral distribution (LSD) of $ \mathbf{C}_{\tau} $, linking it to the transfer function of a univariate linear process.
• Apply tools from random matrix theory, including moment methods and concentration inequalities, to prove convergence of the ESD to the LSD.
• Relax the structural assumptions through discussion of alternative conditions involving limits of traces of polynomials of coefficient matrices, inspired by free probability.

Experimental results

Research questions

• RQ1 Does the empirical spectral distribution of the renormalized and symmetrized sample autocovariance matrix converge almost surely in the moderately high-dimensional regime ($ p/n \to 0 $)?
• RQ2 What is the limiting spectral distribution (LSD) of the matrix $ \mathbf{C}_{\tau} = \sqrt{n/p}(\mathbf{S}_{\tau} - \mathbf{\Sigma}_{\tau}) $, and how is it characterized?
• RQ3 How does the LSD depend on the temporal dependence structure of the linear process, particularly through the transfer function?
• RQ4 Can the results be extended beyond the assumption of simultaneously diagonalizable coefficient matrices?
• RQ5 What are the implications of the LSD for inference on model parameters and prediction in high-dimensional time series?

Key findings

• The empirical spectral distribution of $ \mathbf{C}_{\tau} $ converges almost surely to a nonrandom limit distribution supported on the real line under the given asymptotic regime.
• The limiting spectral distribution is characterized via the Stieltjes transform and depends on the transfer function of the underlying univariate linear process.
• The LSD is non-degenerate and distinct from the semi-circle law, reflecting the impact of temporal dependence in high-dimensional settings.
• The convergence holds under weak regularity conditions: i.i.d. innovations with zero mean, unit variance, and finite fourth moments.
• The structural assumption of Hermitian and simultaneously diagonalizable coefficient matrices can be relaxed to conditions involving limits of averaged traces of polynomials of these matrices.
• The results provide a foundation for analyzing linear spectral statistics and generalizing univariate standard error bounds to high-dimensional time series.