{
  "abstract": "Let \\(\\{X_n:n\\ge0\\}\\) be a controlled branching process with controls \\(ϕ_n\\), and let \\(N_n=ϕ_n(X_n)\\) be its progenitor count. We study large deviations of \\(X_{n+1}/N_n\\), \\(N_n/X_n\\), and \\(X_{n+1}/X_n\\), on their natural positive-denominator events, in supercritical and critical regimes. We treat generation-wide random slopes and asymptotically deterministic first-order slopes. For asymptotically affine random-slope controls, an affine recursion and a random environmental product yield a sharp Bahadur--Rao--Petrov upper-tail asymptotic. A positive Cramér tilt identifies the prefactor via a nontrivial normalized limit, and conditional on the upper-deviation event the three ratios concentrate around their environmental centers. With individual-sum controls, the process is an exact branching process in a random environment with immigration (BPREI). Motivated by this connection, we establish sharp upper large-deviation asymptotics for BPREI and transfer its positive-transform and harmonic-moment profiles to the controlled process. In the nondecreasing supercritical case, harmonic moments exhibit an exact environmental--boundary--persistence trichotomy. In a centered finite-variance critical class, finite-horizon-minimum methods yield exact \\(n^{-1/2}\\)-scale limits for all positive harmonic orders and for the three ratio-deviation probabilities. For deterministic first-order slopes, the environmental-product mechanism disappears. Matched lower and upper probability-generating-function envelopes transfer auxiliary exact-recursion profiles to controls without an exact branching recursion, producing distinct supercritical and critical harmonic-moment trichotomies. Thus the location of first-order randomness determines the lower-tail mechanism, while immigration may change a sharp constant, a phase boundary, or the decay exponent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23696",
  "author_total": 4,
  "authors": [
    {
      "name": "Miguel González"
    },
    {
      "name": "Carmen Minuesa"
    },
    {
      "name": "Inés del Puerto"
    },
    {
      "name": "Anand N. Vidyashankar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0tey91caydfwe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23696",
  "fulltext_url": null,
  "id": "0tey91caydfwe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23696",
  "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"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7214033761",
  "pdf_url": "https://arxiv.org/pdf/2609.23696",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0tey91caydfwe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23696",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23696"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Large Deviations for Controlled Branching Processes: Random Slopes, Harmonic Moments, and Transfer Principles",
  "title_text": "Large Deviations for Controlled Branching Processes: Random Slopes, Harmonic Moments, and Transfer Principles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:33Z",
  "url": "https://research.draconitic.com/paper/0tey91caydfwe",
  "venue": "arXiv",
  "versions": []
}