Tail representations for absolute, signed-power, inverse, and logarithmic moments

Abstract

Let $X$ be a real-valued random variable and let $p>0$. When $X$ can take negative values, $X^p$ is not generally real-valued for non-integer $p$; formulas for moments of arbitrary positive order must therefore distinguish the absolute power $|X|^p$ from the signed power $\operatorname{sgn}(X)|X|^p$. Using the layer-cake identity, we give a self-contained treatment of tail representations for the positive and negative parts of $X$, including absolute and signed-power moments, centered forms, and exact formulas on ordered discrete supports and integer lattices. The basic identities require neither a density nor continuity and apply to continuous, discrete, singular, and mixed distributions. For strictly positive random variables, we also give representations and finiteness criteria for inverse moments, together with distribution-function and Frullani--Laplace formulas for logarithmic moments. Examples illustrate two-sided support, atoms, lattice tails, and behavior near zero. The presentation places these classical identities in a common notation and states the domain and integrability conditions required in each case.