🤖 AI Summary
This work addresses the lack of a unified framework for real-order moments—including positive, fractional, and negative orders—of arbitrary random variables on the real line, encompassing discrete, continuous, and mixed distributions. By extending the tail-integral identity known for non-negative variables, the authors develop a comprehensive representation valid for distributions supported on the entire real axis. The proposed framework expresses real-order moments via integrals of the cumulative distribution and survival functions, and yields series representations based on cumulative probabilities in the discrete case. It further uncovers a geometric–analytic connection between moment existence and tail decay, and establishes links between logarithmic moments, Laplace transforms, and Frullani’s identity. The approach provides concise existence criteria and is validated on Zeta and Skellam distributions, thereby deepening the theoretical interplay among tail probabilities, distribution functions, and moments.
📝 Abstract
This paper develops a unified framework for the study of real-order moments of arbitrary random variables. General integral representations are established in terms of cumulative distribution functions and survival functions, covering continuous, discrete, and mixed distributions supported on the whole real line. These formulas extend the classical tail-integral identities for nonnegative random variables and provide a common treatment of positive, fractional, and negative moments.
For discrete distributions, explicit series representations are derived in terms of cumulative probabilities, yielding simple criteria for the existence of moments. Applications are presented for the zeta and Skellam distributions, illustrating how tail behavior determines moment finiteness and how moments can be represented geometrically through cumulative distribution functions. In addition, a representation for logarithmic moments is obtained, linking logarithmic means, Laplace transforms, and the classical Frullani identity.
The results provide a unified perspective on moment representations and establish useful connections between tail probabilities, distribution functions, Laplace transforms, and moment existence.