{
  "abstract": "Consider a one-dimensional branching Brownian motion. Let $W_\\infty(\\beta)$ denote the limit of the additive martingale in the subcritical regime $\\lvert \\beta\\rvert < \\beta_c$ and $Z_\\infty$ be the limit of the derivative martingale at criticality. Madaule (\\textit{Stochastic Process. Appl.} 126 (2016), no. 2, 470--502) established the following convergence \\[ \\frac{W_\\infty(\\beta)}{\\beta_c-\\beta}\\xrightarrow[\\beta\\nearrow \\beta_c]{\\mathbb{P}} 2Z_\\infty. \\] The goal of this paper is twofold: firstly, we strengthen this result into an almost sure convergence; secondly, we describe the fluctuations occurring in this convergence by proving \\[ \\frac{1}{\\beta_c-\\beta}\\left( \\frac{W_\\infty(\\beta)}{\\beta_c-\\beta} - 2 Z_\\infty +2(\\beta_c-\\beta)\\log(\\beta_c-\\beta) Z_\\infty\\right) \\xrightarrow[\\beta\\nearrow \\beta_c]{(\\mathrm{d})} S, \\] where, conditionally on $Z_\\infty$, $S$ follows a spectrally negative 1-stable distribution with scale and shift parameters proportional to $Z_\\infty$. Furthermore, these results are extended to the setting of complex additive martingales and the fluctuations to a multi-dimensional convergence.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10530",
  "author_total": 2,
  "authors": [
    {
      "name": "Xinxin Chen"
    },
    {
      "name": "Michel Pain"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4w5yqvs9fnyt2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10530",
  "fulltext_url": null,
  "id": "4w5yqvs9fnyt2",
  "issns": [],
  "landing_url": "https://hal.science/hal-05745098v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-mp"
    ],
    [
      "hal",
      "1.math.math-pr"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7212186385",
  "pdf_url": "https://hal.science/hal-05745098/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4w5yqvs9fnyt2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:33Z",
      "native_id": "hal-05745098",
      "source": "hal",
      "url": "https://hal.science/hal-05745098v1"
    },
    {
      "first_seen": "2026-10-30T12:20:12Z",
      "native_id": "2609.10530",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10530"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Fluctuations of additive martingale limits of branching Brownian motion",
  "title_text": "Fluctuations of additive martingale limits of branching Brownian motion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:16Z",
  "url": "https://research.draconitic.com/paper/4w5yqvs9fnyt2",
  "venue": "HAL",
  "versions": []
}