What should we say about the kurtosis?

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Abstract
In this work, we point out some important properties of the normalized fourth-order cumulant (i.e., the kurtosis). In addition, we emphasize the relation between the signal distribution and the sign of the kurtosis. One should mention that in many situations, authors claim that the sign of the kurtosis depends on the nature of the signal (i.e., over- or sub-Gaussian). For a unimodal probability density function, that claim is true and is clearly proved in the letter. But for more complex distributions, it has been shown that the kurtosis sign may change with parameters and does not depend only on the asymptotic behavior of the distributions. Finally, these results give theoretical explanation to techniques, like nonpermanent adaptation, used in nonstationary situations.