What should we say about the kurtosis?

 A. Mansour (1), C. Jutten (2)

(1) BMC Research Center (RIKEN), Moriyama-ku, Nagoya 463 (JAPAN)
(2) INPG - LIS,  46 avenue Felix Viallet, 38031 Grenoble, France


Abstract
In 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.