{
  "abstract": "Let $X_1,X_2,\\ldots$ be i.i.d. complex-valued random variables with arbitrary Borel probability law $μ$, and set $P_n(z)=\\prod_{j=1}^n(z-X_j)$. For every deterministic sequence $k_n=o(n)$, we prove that the empirical zero measure of the $k_n$-th derivative $P_n^{(k_n)}$ converges weakly to $μ$ almost surely.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15984",
  "author_total": 1,
  "authors": [
    {
      "name": "Hongkai Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0ce0qy1396pet.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15984",
  "fulltext_url": null,
  "id": "0ce0qy1396pet",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15984",
  "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",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7213260352",
  "pdf_url": "https://arxiv.org/pdf/2609.15984",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0ce0qy1396pet.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15984",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15984"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Convergence for Small-Order Derivatives of Random Polynomials with Independent Roots",
  "title_text": "Convergence for Small-Order Derivatives of Random Polynomials with Independent Roots",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/0ce0qy1396pet",
  "venue": "arXiv",
  "versions": []
}