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  "slug": "poisson-binomial-density-ratio",
  "title": "A variance bound for Poisson-binomial density ratios",
  "shortTitle": "No outcome of a sum of independent yes/no trials is more likely than under the matching Poisson law by more than the factor mean/variance",
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  "oneLine": "For every count of independent yes/no events with unequal chances, no outcome or event is more likely than under the Poisson law with the same mean by more than the factor mean ÷ variance, proving the 2020 Dümbgen–Wellner conjecture; the factor is not sharp for large means.",
  "abstract": "Counts of independent events with unequal chances have a Poisson-binomial law and are routinely approximated by the Poisson law with the same mean λ. Dümbgen and Wellner (2020) bounded the largest ratio ρ = sup_k P(X = k)/π_λ(k) by 1/(1 − max p_i) and conjectured ρ ≤ λ/v, where v is the variance of the count; they proved it for λ ≤ 1. This candidate proves the conjecture in general (Theorem 1.1): for every sum X of independent Bernoulli variables, finite or countable, optionally plus an independent Poisson part, with mean λ and variance v > 0, and every k, v·P(X = k) ≤ λ·e^(−λ)λ^k/k!. Consequently every event has probability at most λ/v times its Poisson probability, and the Rényi divergence of order infinity is at most log(λ/v). Along two mean-preserving interpolations to the Poisson law, a logistic decay of the odds with Poisson immigration (Theorem 5.1) and the classical thinning semigroup (Theorem 5.2), every variance-weighted atom v(t)P(X_t = k) is nondecreasing. The engine is a family of explicit nonnegative 3 × 3 Toeplitz-minor certificates for convexity of three ratio sequences (Proposition 2.4) with a discrete reverse moment inequality. The same certificates give concavity of ordered conditional inclusion means in rejective sampling (Corollary 7.1). The best constant at fixed Δ = 1 − v/λ lies between max{e^Δ, (1 − Δ)^(−1/2)} and (1 − Δ)^(−1), the bound e^Δ known for small means fails in general (Proposition 8.1), and the inequality does not extend to ultra-log-concave laws (Proposition 8.2). AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.",
  "datePublished": "2026-11-03",
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    "Poisson-binomial distribution",
    "Poisson approximation",
    "density ratio",
    "Dümbgen-Wellner conjecture",
    "Rényi divergence",
    "Toeplitz minors",
    "thinning semigroup",
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    "Theorem 1.1 (proof): for every Poisson–Bernoulli sum X = N + Σ B_i with mean λ and variance v > 0, v·P(X = k) ≤ λ·π_λ(k) for all k; hence ρ ≤ λ/v = (1 − Δ)^(−1), P(X ∈ A) ≤ (λ/v)·P(Poi(λ) ∈ A) and D_∞(L(X) ‖ Poi(λ)) ≤ log(λ/v); strict unless X is Poisson. This is the Dümbgen–Wellner (2020) conjecture, new for λ > 1.",
    "Theorems 5.1 and 5.2 (proof): v(t)·P(X_t = k) is nondecreasing in t along the logistic flow p_i(t) = p_i e^(−t)/(1 − p_i + p_i e^(−t)) with Poisson immigration, and along the thinning semigroup p_i(t) = p_i e^(−t); strictly increasing for finite non-Poisson families.",
    "Proposition 2.4 (proof): explicit sums of nonnegative 3 × 3 Toeplitz minors certify convexity of the reverse ratio b_(k−1)/b_k, the Stein kernel and the forward ratio (k+1)b_(k+1)/b_k.",
    "Corollary 7.1 (proof): in rejective (conditional Poisson) sampling, E[Σ c_i B_i | Σ B_i = k] is concave in k when the weights are ordered like the inclusion probabilities.",
    "Propositions 8.1 and 8.2 (proof): max{e^Δ, (1 − Δ)^(−1/2)} ≤ C(Δ) ≤ (1 − Δ)^(−1), and e^Δ fails for Bin(2, p) with p > log 2; Poisson(1) conditioned on {0, 1, 2, 3} is ultra-log-concave yet violates the bound (ratio ≈ 1.00237)."
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    {
      "citation": "Dümbgen and Wellner (2020). The density ratio of Poisson binomial versus Poisson distributions. Statistics & Probability Letters. Proved the bound 1/(1 − max p_i), proved λ/v (indeed log ρ ≤ Δ) for λ ≤ 1, and conjectured the general bound proved here.",
      "url": "https://doi.org/10.1016/j.spl.2020.108862",
      "doi": "10.1016/j.spl.2020.108862"
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    {
      "citation": "Dümbgen, Samworth and Wellner (2021). Bounding distributional errors via density ratios. Bernoulli. The binomial case of the conjecture and the asymptotics used for the lower bracket in Proposition 8.1.",
      "url": "https://doi.org/10.3150/20-bej1256",
      "doi": "10.3150/20-bej1256"
    },
    {
      "citation": "Hillion and Johnson (2016). Discrete versions of the transport equation and the Shepp–Olkin conjecture. The Annals of Probability. Reverse-ratio convexity and 2 × 2 identities that the 3 × 3 certificates extend; conditional-mean monotonicity.",
      "url": "https://doi.org/10.1214/14-aop973",
      "doi": "10.1214/14-aop973"
    },
    {
      "citation": "Hillion and Johnson (2017). A proof of the Shepp–Olkin entropy concavity conjecture. Bernoulli. Coefficientwise interpolations in the Bernoulli parameters.",
      "url": "https://doi.org/10.3150/16-bej860",
      "doi": "10.3150/16-bej860"
    },
    {
      "citation": "Bonnefont, Hillion and Saumard (2024). Covariance inequalities for convex and log-concave functions. Latin American Journal of Probability and Mathematical Statistics. Lemma 3.1 is a special case of their Theorem 1.2(1), credited.",
      "url": "https://doi.org/10.30757/alea.v21-25",
      "doi": "10.30757/alea.v21-25"
    },
    {
      "citation": "Johnson (2007). Log-concavity and the maximum entropy property of the Poisson distribution. Stochastic Processes and their Applications. The thinning semigroup along which Theorem 5.2 proves monotonicity.",
      "url": "https://doi.org/10.1016/j.spa.2006.10.006",
      "doi": "10.1016/j.spa.2006.10.006"
    },
    {
      "citation": "Barbour and Hall (1984). On the rate of Poisson convergence. Mathematical Proceedings of the Cambridge Philosophical Society. The classical total-variation bound, which remains the better tool for total variation.",
      "url": "https://doi.org/10.1017/s0305004100061806",
      "doi": "10.1017/s0305004100061806"
    },
    {
      "citation": "Hoeffding (1956). On the Distribution of the Number of Successes in Independent Trials. The Annals of Mathematical Statistics. One-sided tail comparisons for Poisson-binomial counts.",
      "url": "https://doi.org/10.1214/aoms/1177728178",
      "doi": "10.1214/aoms/1177728178"
    },
    {
      "citation": "Hough, Krishnapur, Peres and Virág (2006). Determinantal Processes and Independence. Probability Surveys. Counts of determinantal point processes are Bernoulli sums, so the bound applies to them.",
      "url": "https://doi.org/10.1214/154957806000000078",
      "doi": "10.1214/154957806000000078"
    },
    {
      "citation": "Pitman (1997). Probabilistic Bounds on the Coefficients of Polynomials with Only Real Zeros. Journal of Combinatorial Theory, Series A. Real-rooted generating polynomials are exactly Bernoulli-sum laws.",
      "url": "https://doi.org/10.1006/jcta.1997.2747",
      "doi": "10.1006/jcta.1997.2747"
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