{
  "abstract": "We study inference on the drift of a critical Galton--Watson process with immigration, a count time series with a unit root. Climate change motivates such nonstationary models for weather-related disaster counts. The drift is the expected increase in the count per period. Ordinary least squares is inconsistent for the drift, so we study state-weighted least squares, giving greater weight to observations following small counts, where conditional variance is lower. Under strict recurrence, we apply null-recurrent regenerative limit theory to obtain a polynomial convergence rate and a standard normal studentized limit. Our main contribution is the recurrence boundary, where we establish a logarithmic convergence rate and a parameter-free non-Gaussian studentized limit. Estimating the optimal weights yields the same first-order limiting distribution as knowing them. Simulations show lower root mean squared error than time-weighted least squares with weights $1/t$, and improved state-weighted confidence-interval coverage when the unit root is imposed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17999",
  "author_total": 1,
  "authors": [
    {
      "name": "Yang Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7wrn2xz3xfgxr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17999",
  "fulltext_url": null,
  "id": "7wrn2xz3xfgxr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17999",
  "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.ST"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7213527891",
  "pdf_url": "https://arxiv.org/pdf/2609.17999",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7wrn2xz3xfgxr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2609.17999",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17999"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Drift Inference for Unit-Root Galton-Watson Processes with Immigration",
  "title_text": "Drift Inference for Unit-Root Galton-Watson Processes with Immigration",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:03Z",
  "url": "https://research.draconitic.com/paper/7wrn2xz3xfgxr",
  "venue": "arXiv",
  "versions": []
}