Pith Number

pith:DV77N3Z6

pith:2026:DV77N3Z6KGTSQY263GKKNPQRMM

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Sub-Gaussian Concentration and Entropic Normality of the Maximum Likelihood Estimator

Alex Dytso, Leighton P. Barnes

The normalized maximum likelihood estimator satisfies sub-Gaussian tail bounds and converges in relative entropy to a Gaussian under assumptions on the score.

arxiv:2605.07107 v2 · 2026-05-08 · cs.IT · math.IT · math.ST · stat.ML · stat.TH

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Claims

C1strongest claim

With additional assumptions on the score, we first establish sub-Gaussian tail bounds and convergence of all moments for the normalized estimation error. We then prove an entropic central limit theorem for a smoothed version of the estimator, showing convergence in relative entropy to the limiting Gaussian law. When the Fisher information of the normalized estimate is bounded, or its density has bounded first derivative, we further show that the smoothing can be removed, yielding entropic normality of the MLE itself.

C2weakest assumption

Additional assumptions on the score function (beyond standard regularity conditions for the classical CLT), plus bounded Fisher information or bounded first derivative of the density to remove smoothing; these are invoked to obtain the sub-Gaussian and entropic results.

C3one line summary

Under regularity conditions plus assumptions on the score, the normalized MLE has sub-Gaussian tails, all moments converge, and the estimator converges in relative entropy to Gaussian when Fisher information is bounded or the density has bounded derivative.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-26T01:02:35.460660Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

1d7ff6ef3e51a728635ed994a6be116337c4b59d4973c81ae591927112896d12

Aliases

arxiv: 2605.07107 · arxiv_version: 2605.07107v2 · doi: 10.48550/arxiv.2605.07107 · pith_short_12: DV77N3Z6KGTS · pith_short_16: DV77N3Z6KGTSQY26 · pith_short_8: DV77N3Z6

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/DV77N3Z6KGTSQY263GKKNPQRMM \
  | 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: 1d7ff6ef3e51a728635ed994a6be116337c4b59d4973c81ae591927112896d12

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "4e55b01d60d7984c861fde8e37603440842a5053e72ed8344eb54fa49f1a3e69",
    "cross_cats_sorted": [
      "math.IT",
      "math.ST",
      "stat.ML",
      "stat.TH"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.IT",
    "submitted_at": "2026-05-08T01:34:03Z",
    "title_canon_sha256": "2a4a680a09694c437271b2b184fca019a9842b1c3e38d77f082eaaaca661a2b3"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.07107",
    "kind": "arxiv",
    "version": 2
  }
}