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 (yn)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 to the sample covariance matrix of an independent identically NC(0,IM) 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.
Highlights
Under the hypothesis H0 that the M components (y1,n)n∈Z, . . . ,n∈Z of y are mutually uncorrelated, we evaluate the behaviour of certain Linear Spectral Statistics (LSS) of the eigenvalues of C (ν) in asymptotic regimes where N → +∞ and both M = M (N ) and B = B(N ) converge towards +∞ in such a way that
Our main results are obtained using tools borrowed from large random matrix theory and from frequency domain time series analysis techniques
It is still possible to develop large random matrix-based approaches testing that the components of y are uncorrelated or not, see e.g. the contribution [29] to be presented below which, under the extra assumption that the components of y share the same spectral density, is based on a Gaussian approximation of linear spectral statistics of the empirical covariance matrix R N defined by n=1 (1.10)
Summary
We consider an M –variate zero-mean complex Gaussian stationary time series 1 (yn)n∈Z and assume that the samples y1, . . . , yN are available. If (sm)m=1,...,M represent the spectral densities of the scalar time series ((ym,n)n∈Z)m=1,...,M , for each function f defined on R+ and C∞ in a neighbourhood of the support [λ−; λ+] of μ(Mc)P , it holds that for each > 0, there exists a γ( ) := γ > 0 such that for each N large enough: P. and where φN (f ) is a deterministic O(1) term which coincides with the action of the function f on a certain compactly supported distribution DN (to be made precised later) depending on the Marcenko-Pastur distribution μ(McNP). Our approach is based on the observation that in the above asymptotic regime, S(ν) can be interpreted as the sample covariance matrix of the large vectors (ξy(ν b N. zero mean complex random vectors with covariance matrix S(ν) where S(ν) = diag(s1(ν), . Our main results are obtained using tools borrowed from large random matrix theory (see e.g. [30], [1]) and from frequency domain time series analysis techniques (see e.g. [4])
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