{
  "abstract": "Temporal dependence can separate the Gaussian approximation of stochastic gradient descent from its stationary moments. For unmodified least-squares SGD, we construct a design with standard Gaussian marginals whose stationary error has every positive moment infinite. Independent observations with the same marginals instead give finite stationary variance. Both regimes retain a Gaussian small-step limit. Our general theory establishes pathwise contraction from a finite second design moment, then uses score cancellation and localization to obtain stationary Gaussian and Ornstein--Uhlenbeck limits. Independent Gaussian regression errors yield an exact conditional Gaussian law and total-variation convergence under the same design integrability. Stronger design conditions identify a positive first-order total-variation constant and a deterministic covariance correction with $o(a)$ error. A scalar coverage expansion translates this correction into its inference consequence. Experiments examine distributional error, coverage, and calibration with dependent scores. Together, these results establish precise probability-law approximation beyond moment-based stationary analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05599",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaoli Li"
    },
    {
      "name": "Wei Biao Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/93f3mxp0b7efn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05599",
  "fulltext_url": null,
  "id": "93f3mxp0b7efn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05599",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7220739644",
  "pdf_url": "https://arxiv.org/pdf/2610.05599",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/93f3mxp0b7efn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:16Z",
      "native_id": "2610.05599",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05599"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Gaussian Limits for SGD Without Stationary Moments",
  "title_text": "Gaussian Limits for SGD Without Stationary Moments",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:16Z",
  "url": "https://research.draconitic.com/paper/93f3mxp0b7efn",
  "venue": "arXiv",
  "versions": []
}