Properties of linear spectral statistics of frequency-smoothed estimated spectral coherence matrix of high-dimensional Gaussian time series

TL;DR

This paper analyzes the asymptotic properties of linear spectral statistics of frequency-smoothed spectral coherence matrices in high-dimensional Gaussian time series, showing convergence to the Marcenko-Pastur distribution and providing explicit deterministic approximations.

Contribution

It provides a rigorous analysis of the spectral behavior of smoothed spectral coherence matrices in high dimensions, including explicit formulas and concentration bounds.

Findings

Eigenvalue distribution converges to Marcenko-Pastur law

Spectral statistics have explicit deterministic limits

Deviation bounds depend on sample size and dimension

Abstract

The asymptotic behaviour of Linear Spectral Statistics (LSS) of the smoothed periodogram estimator of the spectral coherency matrix of a complex Gaussian high-dimensional time series $(\y_n)_{n \in \mathbb{Z}}$ with independent components is studied under the asymptotic regime where the sample size $N$ converges towards $+\infty$ while the dimension $M$ of $\y$ and the smoothing span of the estimator grow to infinity at the same rate in such a way that $\frac{M}{N} \rightarrow 0$. It is established that, at each frequency, the estimated spectral coherency matrix is close from the sample covariance matrix of an independent identically $\mathcal{N}_{\mathbb{C}}(0,\I_M)$ distributed sequence, and that its empirical eigenvalue distribution converges towards the Marcenko-Pastur distribution. This allows to conclude that each LSS has a deterministic behaviour that can be evaluated explicitly. Using concentration inequalities, it is shown that the order of magnitude of the supremum over the frequencies of the deviation of each LSS from its deterministic approximation is of the order of $\frac{1}{M} + \frac{\sqrt{M}}{N}+ (\frac{M}{N})^{3}$ where $N$ is the sample size. Numerical simulations supports our results.