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, Xp 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 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.
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BibTeX
@misc{vila2026tail,
title = {{Tail representations for absolute, signed-power, inverse, and logarithmic moments}},
author = {Roberto Vila and Eduardo Nakano and Cecilia Castro},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.14019},
eprint = {2606.14019},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.14019},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Tail representations for absolute, signed-power, inverse, and logarithmic moments
AU - Roberto Vila
AU - Eduardo Nakano
AU - Cecilia Castro
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2606.14019
UR - https://arxiv.org/abs/2606.14019
AB - 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 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.
ER -