{
  "abstract": "We study randomly weighted sums \\[ T_n=\\frac{\\sum_{i=1}^n X_iY_i}{\\sum_{i=1}^n Y_i}, \\] where $(X_i)$ and $(Y_i)$ are independent sequences of i.i.d. random variables, with nonnegative weights $Y_i$, motivated by Breiman's conjecture. We introduce the class $ISSV$ of $I$-subsequentially slowly varying tails and establish a subsequential analogue of Darling's maximum--sum theorem. This provides a subsequential generalization of Breiman's slowly varying theorem and, as a corollary, resolves the slowly varying case of his conjecture. We also establish a subsequential analogue of Breiman's nonzero-index theorem, characterizing the existence of nondegenerate subsequential limits, distinct in distribution from $X_1,$ by the failure of slow variation of the integrated tail, for weight $Y_1$ with tail distributions outside the class $ISSV$. Finally, we formulate the subsequential separation principle, prove it under a finite-second-moment assumption, and show that its general validity would imply the nonzero-index case of Breiman's conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29551",
  "author_total": 1,
  "authors": [
    {
      "name": "Volodymyr Pavlenkov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/42xvbwce9gmq8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29551",
  "fulltext_url": null,
  "id": "42xvbwce9gmq8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29551",
  "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",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7223220390",
  "pdf_url": "https://arxiv.org/pdf/2610.29551",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/42xvbwce9gmq8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29551",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29551"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Subsequential Version of Breiman's Conjecture",
  "title_text": "Subsequential Version of Breiman's Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/42xvbwce9gmq8",
  "venue": "arXiv",
  "versions": []
}