What should we say about the kurtosis ?

International audienceIn this paper 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 unimodal probability density function, that claim is true and is clearly proved in this paper. However, 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 provide a theoretical explanation to techniques, like non-permanent adaptation, used in nonstationary situations