{
  "abstract": "Expectiles are tail-sensitive distributional summaries with important applications in financial risk measurement and forecasting, yet statistical theory for their online estimation from dependent data remains limited. We study a one-pass, stochastic-gradient algorithm with constant learning rate $γ$ for estimating the marginal expectile of a stationary causal process. The recursion admits a unique stationary causal solution and geometrically forgets its initialization. Under short-range dependence, its last iterate is asymptotically Gaussian at the $\\sqrtγ$ scale, with variance governed by the population curvature and the long-run variance of the expectile score. For long-memory linear processes the scale changes to $γ^{β-1/2}\\ell(1/γ)$, with a Gaussian limit arising from a fractional Brownian motion integral. We further derive the joint long-memory limit for multiple expectile levels and characterize its limiting covariance structure. Various numerical experiments corroborate the predicted Gaussian approximations and regime-specific fluctuation scales.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16030",
  "author_total": 3,
  "authors": [
    {
      "name": "Younghun Lee"
    },
    {
      "name": "Jake Zummo"
    },
    {
      "name": "Wei Biao Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2wkmb5yq7fb76.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16030",
  "fulltext_url": null,
  "id": "2wkmb5yq7fb76",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16030",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7221758006",
  "pdf_url": "https://arxiv.org/pdf/2610.16030",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2wkmb5yq7fb76.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16030",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16030"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Limit Theory for Online Expectile Estimation under Short and Long Range Dependence",
  "title_text": "Limit Theory for Online Expectile Estimation under Short and Long Range Dependence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/2wkmb5yq7fb76",
  "venue": "arXiv",
  "versions": []
}