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Journal Articles Random Matrices: Theory and Applications Year : 2022

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

Abstract

We consider linear spectral statistics built from the block-normalized correlation matrix of a set of M mutually independent scalar time series. This matrix is composed of M² blocks. Each block has size L×L and contains the sample cross-correlation measured at L consecutive time lags between each pair of time series. Let N denote the total number of consecutively observed windows that are used to estimate these correlation matrices. We analyze the asymptotic regime where M, L, N → +∞ while ML/N → c_k, 0 < c_k < ∞. We study the behavior of linear statistics of the eigenvalues of this block correlation matrix under these asymptotic conditions and show that the empirical eigenvalue distribution converges to a Marcenko–Pastur distribution. Our results are potentially useful in order to address the problem of testing whether a large number of time series are uncorrelated or not.
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Dates and versions

hal-02543994 , version 1 (15-04-2020)
hal-02543994 , version 2 (14-01-2021)

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Philippe Loubaton, Xavier Mestre. On the asymptotic behavior of the eigenvalue distribution of block correlation matrices of high-dimensional time series. Random Matrices: Theory and Applications, 2022, 11 (31), pp.2250024. ⟨10.1142/S2010326322500241⟩. ⟨hal-02543994v2⟩
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