{
  "abstract": "We study the optimal score $T_{n,m}$ in a complete feedback card guessing game. The deck has $n$ types and $m$ cards of each type. Its predictable compensator $S_{n,m}$ is the sum of the conditional success probabilities. We prove that Var$(S_{n,m})\\leq C((\\log\\log n)^2+m/\\log n)$, uniformly in $m$ for large $n$. This bound yields a central limit theorem for every sequence $m=m_n\\geq1$ as $n\\to\\infty$. The score variance is asymptotic to its mean. We also give uniform Kolmogorov bounds using the conditional Bernoulli moments. For $m,n\\to\\infty$, the excess mean is asymptotic to $(\\log n)\\log\\min\\{m,\\log n\\}+(π/\\sqrt2)\\sqrt{m\\log n}$. The two contributions balance at $m\\asymp(\\log n)(\\log\\log n)^2$. The variance proof uses reversed occupancy hitting times. Discrete hypergeometric tail estimates control their fluctuations in the large-deviation, Gaussian, and saturation regimes. For fixed $n$, we also show that the compensator has a Brownian-bridge functional limit as $m\\to\\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17479",
  "author_total": 2,
  "authors": [
    {
      "name": "Partha S. Dey"
    },
    {
      "name": "Jaya M. Manthripragada"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8sfp017v1vrc4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17479",
  "fulltext_url": null,
  "id": "8sfp017v1vrc4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17479",
  "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|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7221832764",
  "pdf_url": "https://arxiv.org/pdf/2610.17479",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8sfp017v1vrc4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17479",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17479"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Limit Theorems for the Complete Feedback Card Guessing Game",
  "title_text": "Limit Theorems for the Complete Feedback Card Guessing Game",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/8sfp017v1vrc4",
  "venue": "arXiv",
  "versions": []
}