{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.14019",
  "author_total": 3,
  "authors": [
    {
      "name": "Roberto Vila"
    },
    {
      "name": "Eduardo Nakano"
    },
    {
      "name": "Cecilia Castro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4dzfxckpmxf29.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.14019",
  "fulltext_url": null,
  "id": "4dzfxckpmxf29",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.14019",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7164844618",
  "pdf_url": "https://arxiv.org/pdf/2606.14019",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4dzfxckpmxf29.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2606.14019",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.14019"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Tail representations for absolute, signed-power, inverse, and logarithmic moments",
  "title_text": "Tail representations for absolute, signed-power, inverse, and logarithmic moments",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:26Z",
  "url": "https://research.draconitic.com/paper/4dzfxckpmxf29",
  "venue": "arXiv",
  "versions": []
}