Pith Number

pith:JMZMDBMV

pith:2026:JMZMDBMVXDNLVKYVKV537GDC7W

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Distributional Statistical Models: Weak Moments, Cumulants, and a Central Limit Theorem

R. Labouriau

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.

arxiv:2604.20634 v2 · 2026-04-22 · math.PR · math.ST · stat.TH

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

Cited by

4 papers in Pith

Inference Functionals and Observation Operators for Distributional Statistical Models

Notes on Transversality and Statistical Degeneracies in Distributional Models

Weak Moment Methods for Statistical Inference: with an Application to Robust Estimation

Transversality and Geometric Regularisation in Distributional Statistical Models

Receipt and verification

First computed	2026-05-22T01:04:02.864057Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

4b32c18595b8dabaab15557bbf9862fdbd25c8df4d94b7d89d582de07486a300

Aliases

arxiv: 2604.20634 · arxiv_version: 2604.20634v2 · doi: 10.48550/arxiv.2604.20634 · pith_short_12: JMZMDBMVXDNL · pith_short_16: JMZMDBMVXDNLVKYV · pith_short_8: JMZMDBMV

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/JMZMDBMVXDNLVKYVKV537GDC7W \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 4b32c18595b8dabaab15557bbf9862fdbd25c8df4d94b7d89d582de07486a300

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "82f97059b3b7f65986766a82039963cb3f060d8649c768561f003954166495e1",
    "cross_cats_sorted": [
      "math.ST",
      "stat.TH"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-04-22T14:46:39Z",
    "title_canon_sha256": "5bdf8c6407e31bcceecae380b62a8776b301a44c2a4ff739fb97550b718ca24f"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.20634",
    "kind": "arxiv",
    "version": 2
  }
}