A zero-cumulant random variable and its applications

High Order Statistics (HOS) are widely used in many algorithms ranging from blind identification to signal separation. A well-known identifiability result is that at most one Gaussian source should be present in static mixtures [J. Eriksson, V. Koivunen, Identifiability, separability and uniqueness of linear ICA models, IEEE Signal Process. Lett. (2004) 601–604]. The reason for this is that these algorithms utilize cumulants of order higher than two, and that they are all null for circular Gaussian random variables [M. Kendall, A. Stuart, The Advanced Theory of Statistics, Distribution Theory, vol. 1, C. Griffin, 1977]. Simple examples of non-Gaussian complex random variables having zero cumulants of order three to seven are given, which can be encountered in the real world.