The Muculants - a Higher-Order Statistical Approach.

An alternative parametric description for discrete random variables, called muculants, is proposed. Contrary to cumulants, muculants are based on the Fourier series expansion rather than on the Taylor series expansion of the logarithm of the characteristic function. Utilizing results from cepstral theory, elementary properties of muculants are derived. A connection between muculants and cumulants is developed, and the muculants of selected discrete random variables are presented. Specifically, it is shown that the Poisson distribution is the only discrete distribution where only the first two muculants are non-zero, thus being the muculant-counterpart of the simple cumulant structure of Gaussian distributions.