{
  "abstract": "Let $X$ be a continuous Gaussian process in $\\mathbb{R}^d$, $d\\ge2$, with independent components of roughness $H\\in(\\tfrac14,1)$, possibly with a linear drift, and let $X^n$ be its piecewise-linear interpolant on a mesh of width $δ$. We study how well the truncated signature of $X^n$ approximates that of the natural lift of $X$, in mean (the weak error) and in $L^2$ (the strong error). Over a class of drivers defined by quantitative regularity conditions, the weak error at every fixed truncation level is $O(δ^{γ(H)})$ with $γ(H)=\\min(2H,4H-1)$, uniformly over the class, and for truncation levels $M\\ge4$ fractional Brownian motion attains this rate. For centred drivers the weak error vanishes at levels $1$ to $3$, Gaussian or not. It first appears at level~$4$, where we compute its leading constant for independent fractional Brownian motion in $d=2$: on $(\\tfrac12,1)$ it is $1/(2\\sqrt3)$, independent of $H$, and on $(\\tfrac14,\\tfrac12)$ it is given by a $ζ$-series. For fractional Brownian motion with drift, a level-$3$ error of the same order appears for $H\\ge\\tfrac12$. Below $\\tfrac12$ the drift reaches the rate only at levels $\\ge5$ (proved for $d\\ge3$). The strong error is of exact order $δ^{2H-1/2}$ for fractional Brownian motion with $H\\in(\\tfrac12,1)$, so in mean square the weak error lies a full power of $δ$ below it. For $H\\le\\tfrac12$ the strong error is $O(δ^{\\vartheta})$ for every $\\vartheta<2H-\\tfrac12$, uniformly over the class, and the exponent $2H-\\tfrac12$ cannot be improved. Finally, we construct pairs of laws in the class whose mesh data have the same law but whose expected signatures differ by order $δ^{γ(H)}$. Hence no quantity computed from the mesh data has mean within $o(δ^{γ(H)})$ of the expected signature uniformly over the class.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08588",
  "author_total": 1,
  "authors": [
    {
      "name": "Florian Huber"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ed2gj41fh8fxx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08588",
  "fulltext_url": null,
  "id": "ed2gj41fh8fxx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08588",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08588",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ed2gj41fh8fxx.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:07Z",
      "native_id": "2611.08588",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08588"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Weak and strong discretization errors for signatures of Gaussian processes",
  "title_text": "Weak and strong discretization errors for signatures of Gaussian processes",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:07Z",
  "url": "https://research.draconitic.com/paper/ed2gj41fh8fxx",
  "venue": "arXiv",
  "versions": []
}