{
  "abstract": "For an AR($1$) sequence with coefficient $0 < a < 1$, the probability of staying above zero decays exponentially in the number of steps; at the unit root $a=1$ it decays polynomially, with order $n^{-1/2}$ for symmetric increments. We study this transition for AR($1$) sequences driven by symmetric and asymmetric $α$-stable innovations. We write $Λ(a,α)$ for the symmetric exponential persistence rate and $Λ(a,α,ρ)$ for the rate of a strictly stable law with positivity parameter $ρ$. The entire chain admits an exact representation through a single stable Lévy process observed on a geometrically expanding time grid. In the symmetric case, comparison with continuous half-line survival gives $Λ(a,α) \\leq \\fracα{2}\\log{(1/a)}$. For $0 < α< 2$, this bound disproves the stable specialization of a conjecture of Hinrichs, Kolb and Wachtel concerning regularly varying innovation tails. Combining stable closure under subsampling with a monotonicity coupling yields a lower bound of the same near-unit order. This proves $Λ(a,α) \\asymp \\log{(1/a)}$ as $a \\uparrow 1$ and shows that $Λ(a,α)/\\log{(1/a)}$ converges to a limit in $(0,α/2]$ equal to the supremum of that ratio over $0 < a < 1$. A Lamperti transformation reduces identification of this constant to a dense-sampling problem for a stationary stable Ornstein-Uhlenbeck process. Existing Gaussian theory determines the sharp value at $α=2$; for $0 < α< 2$, rescued crossings between observations remain the obstacle. For asymmetric strictly stable innovations with $α\\neq 1$ and admissible $ρ\\in(0,1)$, the corresponding bounds are $Λ(a,α,ρ) \\leq α(1-ρ)\\log{(1/a)}$ and $Λ(a,α,ρ) \\asymp \\log{(1/a)}$. At $α=2$, only the symmetric case $ρ=1/2$ occurs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.17927",
  "author_total": 2,
  "authors": [
    {
      "name": "José Ricardo G. Mendonça"
    },
    {
      "name": "Boubaker Smii"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2k702181k3k8w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.17927",
  "fulltext_url": null,
  "id": "2k702181k3k8w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.17927",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7203779589",
  "pdf_url": "https://arxiv.org/pdf/2608.17927",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2k702181k3k8w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:12Z",
      "native_id": "2608.17927",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.17927"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Near-unit-root persistence of symmetric and asymmetric stable autoregressive sequences",
  "title_text": "Near-unit-root persistence of symmetric and asymmetric stable autoregressive sequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:26Z",
  "url": "https://research.draconitic.com/paper/2k702181k3k8w",
  "venue": "arXiv",
  "versions": []
}