{
  "abstract": "Let $(X_1,\\ldots,X_n)$ be a random variable in $(\\mathbb{R}\\setminus\\{0\\})^n$ that is both exchangeable and sign-invariant. For every $k\\in[n]$, let $S_k=\\sum_{i=1}^kX_i$. Define the weak persistence probability as $\\mathbb{P}(S_1,\\ldots,S_n\\geq0)$, and the strong persistence probability as $\\mathbb{P}(S_1,\\ldots,S_n>0)$. From previous results, the optimal lower bound for $\\mathbb{P}(S_1,\\ldots,S_n\\geq0)$ and the optimal upper bound for $\\mathbb{P}(S_1,\\ldots,S_n>0)$ are known. We complete the picture by determining the optimal upper bound for $\\mathbb{P}(S_1,\\ldots,S_n\\geq0)$ and the optimal lower bound for $\\mathbb{P}(S_1,\\ldots,S_n>0)$ as follows. \\[\\frac{1}{2^n}\\binom{n-1}{\\lfloor (n-1)/2\\rfloor}\\leq\\mathbb{P}(S_1,\\ldots,S_n>0)\\leq\\frac{1}{4^n}\\binom{2n}{n}\\leq\\mathbb{P}(S_1,\\ldots,S_n\\geq0)\\leq\\frac{1}{2^n}\\binom{n}{\\lfloor n/2\\rfloor}.\\] In particular, this implies that the weak and strong persistence probabilities of every exchangeable and sign-invariant random variable $(X_1,\\ldots,X_n)$ in $(\\mathbb{R}\\setminus\\{0\\})^n$ are of the order $n^{-1/2}$. We also obtain some related results, one in the deterministic setting, and one when the random variables are allowed to take the value 0.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05586",
  "author_total": 2,
  "authors": [
    {
      "name": "Daniel Iľkovič"
    },
    {
      "name": "Jun Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bjgc4zgbgcwwa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05586",
  "fulltext_url": null,
  "id": "bjgc4zgbgcwwa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05586",
  "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.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7212070944",
  "pdf_url": "https://arxiv.org/pdf/2609.05586",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bjgc4zgbgcwwa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05586",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05586"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Extremal persistence probabilities of exchangeable sign-invariant random variables",
  "title_text": "Extremal persistence probabilities of exchangeable sign-invariant random variables",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/bjgc4zgbgcwwa",
  "venue": "arXiv",
  "versions": []
}