Cepstral Analysis of Random Variables: Muculants

TL;DR

This paper introduces muculants, a new parametric method based on Fourier series for describing discrete random variables, offering advantages over cumulants in terms of existence and convergence properties.

Contribution

The paper proposes muculants as an alternative to cumulants, deriving their properties, and demonstrating their advantages through theoretical analysis and examples.

Findings

Muculants are additive like cumulants.

All muculants exist under Paley-Wiener condition.

Poisson distribution uniquely has only two nonzero muculants.

Abstract

An alternative parametric description for discrete random variables, called muculants, is proposed. In contrast to cumulants, muculants are based on the Fourier series expansion, rather than on the Taylor series expansion, of the logarithm of the characteristic function. We utilize results from cepstral theory to derive elementary properties of muculants, some of which demonstrate behavior superior to those of cumulants. For example, muculants and cumulants are both additive. While the existence of cumulants is linked to how often the characteristic function is differentiable, all muculants exist if the characteristic function satisfies a Paley-Wiener condition. Moreover, the muculant sequence and, if the random variable has finite expectation, the reconstruction of the characteristic function from its muculants converge. We furthermore develop a connection between muculants and cumulants and present the muculants of selected discrete random variables. Specifically, it is shown that the Poisson distribution is the only distribution where only the first two muculants are nonzero.