{
  "abstract": "Let $P$ be an irreducible reversible Markov kernel on an $m$-state space $Ω$, and denote its right spectral gap by $\\Gap(P) =1-λ_2(P)$. For a stationary trajectory, let $K_t$ be the occupation-count vector of the length-$n$ window beginning at time $t$. The stationary pair $(K_0,K_1)$ defines a reversible projected count kernel $\\widetilde P_n$. For each fixed $m\\ge2$, define \\[ c_m^\\star= \\inf_{\\substack{|Ω|=m,\\ P\\ \\mathrm{irreducible\\ and\\ reversible} n\\ge2}} \\frac{n\\Gap(\\widetilde P_n)}{\\Gap(P)}. \\] We prove \\[ \\frac1{1080m}\\le c_m^\\star\\le q_{m-2}, \\qquad q_0=\\frac14,\\quad q_{r+1}=q_r(1-q_r). \\] We also show $q_{m-2}=(m+\\log m+O(1))^{-1}$, which implies that $c_m^\\star=Θ(m^{-1})$. Thus, the reciprocal optimal comparison coefficient $C_m^{\\mathrm{opt}}=(c_m^\\star)^{-1}$ is of the order $m$. In the two-state case, we determine the exact values $c_2^\\star=1/4$ and $C_2^{\\mathrm{opt}}=4$. %the exact constants for $m\\ge3$ remain open. The lower bound also holds for $n=1$ and is uniform in $P$, including sparse and periodic kernels. The proof combines short-window decorrelation with an averaged anchor-excursion decomposition, a Green-kernel hitting estimate, and a stopped Carleson--Hardy inequality. A nested rare-state construction produces finite $m$-state witnesses whose normalized Rayleigh quotients approach $q_{m-2}$ through an ordered sequence of limits. For every fixed finite irreducible reversible aperiodic kernel on at least two states, $\\Gap(\\widetilde P_n)=Θ_P(n^{-1})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27836",
  "author_total": 2,
  "authors": [
    {
      "name": "Yanjin Xiang"
    },
    {
      "name": "Zhihua Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8m9kkkayg451b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27836",
  "fulltext_url": null,
  "id": "8m9kkkayg451b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27836",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Markov Chains and Monte Carlo Methods"
    ]
  ],
  "openalex": "W7214159263",
  "pdf_url": "https://arxiv.org/pdf/2609.27836",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8m9kkkayg451b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2609.27836",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27836"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Poincaré's Inequality on Sliding-Window Occupation Count Kernels and the Optimal Normalized Spectral-Gap Constant",
  "title_text": "Poincaré's Inequality on Sliding-Window Occupation Count Kernels and the Optimal Normalized Spectral-Gap Constant",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:00Z",
  "url": "https://research.draconitic.com/paper/8m9kkkayg451b",
  "venue": "arXiv",
  "versions": []
}