Large random matrix approach for testing independence of a large number of Gaussian time series

2020/07/17 by Philippe Loubaton, Loubaton, Philippe, Alexis Rosuel +1
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Probability (math.PR) #Random Matrices and Applications #Statistical Methods and Bayesian Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2007.08806

openalex publication_date 2020/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∈ ℤ} with independent components is studied under the asymptotic regime where the sample size N converges towards +∞ 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 (M)/(N) → 0. It is established that, at each frequency, the estimated spectral coherency matrix is close from the sample covariance matrix of an independent identically N_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 (1)/(M) + (√(M))/(N)+ ((M)/(N))³ where N is the sample size. Numerical simulations supports our results.

Cited by

• On the asymptotic behaviour of the eigenvalue distribution of block correlation matrices of high-dimensional time series

Related

• Properties of linear spectral statistics of frequency-smoothed estimated spectral coherence matrix of high-dimensional Gaussian time series
• Correlation tests and sample spectral coherence matrix in the high-dimensional regime
• On the asymptotic distribution of the maximum sample spectral coherence of Gaussian time series in the high dimensional regime
• On the frequency domain detection of high dimensional time series
• Asymptotics of empirical eigenvalues for large separable covariance matrices