{
  "abstract": "We establish a functional central limit theorem for locally stationary time series in a separable Banach space $B$. The result requires no type-2 or cotype assumptions and thus covers spaces central to functional data analysis, such as $C([0,1])$ and $L^p([0,1])$. Under moment conditions, summable physical dependence coefficients and a bracketing entropy condition controlling the infinite-dimensional tails, the centered partial sum process converges weakly in $D([0,1],B)$ to a centered Gaussian process, whose covariance is the integrated local long-run covariance, an element of the projective tensor product. We also obtain a stochastic integral representation with respect to a cylindrical Brownian motion. As the limit depends on the unknown long-run covariance, we prove consistency of a block multiplier bootstrap for the partial sum process. This yields CUSUM tests for a constant mean of the full process, based on arbitrary continuous seminorms and valid under a time-varying long-run covariance. For linear projections, we derive a self-normalized test with a pivotal limit. In simulations, the supremum norm test detects localized changes that its $L^2$ counterpart largely misses, and applications to EEG and temperature data illustrate the methodology. Examples in $C([0,1])$ and $W^{1,p}_0([0,1])$ show how the entropy condition can be verified.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25791",
  "author_total": 2,
  "authors": [
    {
      "name": "Florian Heinrichs"
    },
    {
      "name": "Luis-Alberto Rodríguez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6nzba6fqsekeq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25791",
  "fulltext_url": null,
  "id": "6nzba6fqsekeq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25791",
  "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",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Advanced Statistical Methods and Models"
    ]
  ],
  "openalex": "W7214111107",
  "pdf_url": "https://arxiv.org/pdf/2609.25791",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6nzba6fqsekeq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2609.25791",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25791"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "A Functional Central Limit Theorem for Locally Stationary Time Series in Banach Spaces",
  "title_text": "A Functional Central Limit Theorem for Locally Stationary Time Series in Banach Spaces",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:00Z",
  "url": "https://research.draconitic.com/paper/6nzba6fqsekeq",
  "venue": "arXiv",
  "versions": []
}