{"paper":{"title":"Distributional Statistical Models: Weak Moments, Cumulants, and a Central Limit Theorem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A framework using tempered distributions and Schwartz kernels defines weak moments and cumulants that always exist, supporting a central limit theorem for models where classical moments fail.","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"R. Labouriau","submitted_at":"2026-04-22T14:46:39Z","abstract_excerpt":"Many important statistical models fall outside classical moment-based methods due to the non-existence of moments or moment generating functions. We propose a generalised probabilistic framework in which densities are replaced by pairs $(T,\\varphi)$, where $T \\in \\mathcal{S}'(\\mathbb{R})$ is a tempered distribution and $\\varphi \\in \\mathcal{S}(\\mathbb{R})$ is a Schwartz kernel. Expectations are defined via the action of distributions on regularised test functions, yielding well-defined weak moments, weak characteristic functions, and weak cumulants of all orders. These extend classical quantit"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The main results are: (i) a systematic algebra of weak cumulants; (ii) a weak moment problem where existence of all moments holds unconditionally and uniqueness depends on the kernel, with uniqueness results under Gaussian kernels (via Hermite completeness), positive Schwartz kernels with square-integrable densities (via a Carleman-type criterion), and kernels with exponential decay (via Denjoy-Carleman quasi-analyticity); and (iii) a weak central limit theorem formulated as convergence of weak characteristic functions to a Gaussian limit, covering cases where the classical theorem fails. As a statistical consequence, the weak first moment yields a consistent estimator of the location parameter in the Cauchy model.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The framework assumes that the tempered distribution T and Schwartz kernel phi can be chosen so that the weak moments and cumulants retain the algebraic properties of classical ones and that the specific kernels (Gaussian, positive square-integrable, exponentially decaying) deliver the claimed uniqueness via Hermite completeness, Carleman criterion, or Denjoy-Carleman quasi-analyticity.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A distributional framework using tempered distributions and Schwartz kernels defines weak moments and cumulants, supports a weak central limit theorem, and gives consistent location estimation for the Cauchy distribution.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A framework using tempered distributions and Schwartz kernels defines weak moments and cumulants that always exist, supporting a central limit theorem for models where classical moments fail.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"57ea4779d98e16216f6a482d1c592ba49e8fa48a2ce899f386a7b0799aafc544"},"source":{"id":"2604.20634","kind":"arxiv","version":2},"verdict":{"id":"5f8e4c2a-f1b6-46e1-badb-57f2186907d8","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-09T23:11:25.814371Z","strongest_claim":"The main results are: (i) a systematic algebra of weak cumulants; (ii) a weak moment problem where existence of all moments holds unconditionally and uniqueness depends on the kernel, with uniqueness results under Gaussian kernels (via Hermite completeness), positive Schwartz kernels with square-integrable densities (via a Carleman-type criterion), and kernels with exponential decay (via Denjoy-Carleman quasi-analyticity); and (iii) a weak central limit theorem formulated as convergence of weak characteristic functions to a Gaussian limit, covering cases where the classical theorem fails. As a statistical consequence, the weak first moment yields a consistent estimator of the location parameter in the Cauchy model.","one_line_summary":"A distributional framework using tempered distributions and Schwartz kernels defines weak moments and cumulants, supports a weak central limit theorem, and gives consistent location estimation for the Cauchy distribution.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The framework assumes that the tempered distribution T and Schwartz kernel phi can be chosen so that the weak moments and cumulants retain the algebraic properties of classical ones and that the specific kernels (Gaussian, positive square-integrable, exponentially decaying) deliver the claimed uniqueness via Hermite completeness, Carleman criterion, or Denjoy-Carleman quasi-analyticity.","pith_extraction_headline":"A framework using tempered distributions and Schwartz kernels defines weak moments and cumulants that always exist, supporting a central limit theorem for models where classical moments fail."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.20634/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T14:35:48.599013Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-20T01:41:21.478890Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"a816804536ddbfb5e50b43ee35ff508d42bfe59c1b80f142500c0634390dd937"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}