{
  "abstract": "We study prophet inequalities for a random walk reward stopping problem with sample-based information. The goal is to stop as close as possible to the maximum of a random walk with i.i.d. increments, measuring performance by the ratio between the expected reward when stopping and the expected true maximum. We consider a sample-based model in which the increment distribution is unknown and the decision maker has access to $K$ independent sample paths of the reward process. For the infinite-horizon setting, we establish a sharp prophet inequality with constant $(K/(K+1))^{K+1}$. The guarantee is attained by a randomised stopping rule based on the ladder height decomposition of random walks. As $K\\to\\infty$, this recovers the classical $1/e$ prophet inequality from the full-information setting. For the finite-horizon setting, where the process terminates after $n$ steps, we first prove a tight no-information prophet inequality with constant $1/H_n$, where $H_n$ is the $n$-th harmonic number. For $K\\ge1$ samples, we show that a prophet constant of $1/4$ is attainable. Finally, we prove that, with $K$ samples, the prophet constant is at most $(K/(K+1))^{K+1}+(6+6H_K)/H_n$ for $n\\ge 2K^2$, implying convergence to the infinite-horizon constant as $n\\to\\infty$. Our approach combines random walk theory, including ladder heights and Spitzer's identity, with linear programming duality. Our results contribute to random walk stopping theory and to sample-based prophet inequalities for correlated rewards, an area that remains largely unexplored. To the best of our knowledge, our tight sample-based prophet inequalities are the first whose performance is parameterised exactly, rather than only up to constants, by the number of available samples.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26017",
  "author_total": 3,
  "authors": [
    {
      "name": "Pieter Kleer"
    },
    {
      "name": "Johan van Leeuwaarden"
    },
    {
      "name": "Daan Noordenbos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2ygzvwstfxemm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26017",
  "fulltext_url": null,
  "id": "2ygzvwstfxemm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26017",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1705|Optimization and Search Problems"
    ]
  ],
  "openalex": "W7214115230",
  "pdf_url": "https://arxiv.org/pdf/2609.26017",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2ygzvwstfxemm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26017",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26017"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Sample-Based Prophet Inequalities for Random Walks",
  "title_text": "Sample-Based Prophet Inequalities for Random Walks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/2ygzvwstfxemm",
  "venue": "arXiv",
  "versions": []
}