On the Behaviour of the Estimated Fourth-Order Cumulants Matrix of a High-Dimensional Gaussian White Noise

This paper is devoted to the study of the traditional estimator of the fourth-order cumulants matrix of a high-dimensional multivariate Gaussian white noise. If M represents the dimension of the noise and N the number of available observations, it is first established that this M × M matrix converges towards 0 in the spectral norm sens provided M 2 logN N → 0. The behaviour of the estimated fourth-order cumulants matrix is then evaluated in the asymptotic regime where M and N converge towards +∞ in such a way that M 2 N converges towards a constant. In this context, it is proved that the matrix does not converge towards 0 in the spectral norm sense, and that its empirical eigenvalue distribution converges towards a shifted Marcenko-Pastur distribution. It is finally claimed that the largest and the smallest eigenvalue of the cumulant matrix converges almost surely towards the rightend and the leftend points of the support of the Marcenko-Pastur distribution.