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The Poisson binomial distribution -- Old & New

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Pith's one-line read For $X \sim \mathrm{Bin}(3n,1/2)$, the probability generating function of $\lfloor 2X/3 \rfloor$ has a root with imaginary part at least $\sqrt{9n^2-9n-1}/2$, so naive rational rounding does not preserve the strongly Rayleigh property.

desk verdict Useful survey of Poisson binomial theory with a few new results that are under-proved; Theorem 4.4 needs a real proof or a demotion to conjecture. read the letter →

arxiv 1908.10024 v1 pith:CPNKKQIK submitted 2019-08-27 math.PR stat.OT

Wenpin Tang , Fengmin Tang This is my paper · ORCID
classification math.PRstat.OT MSC 60E0560E1060E1562E17
keywords PoissonbinomialdistributionstronglyRayleighpropertyreal-rootedpolynomialsrationalroundingofrandomvariablesoptimaltransportlearningNewton'sinequality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper is an expository survey of the Poisson binomial distribution with a new counterexample at its core. Its unifying thesis is that a distribution on $\{0,\dots,n\}$ is Poisson binomial exactly when its generating polynomial has only real roots, a property called strongly Rayleigh; under this identification, stochastic orderings, Poisson and normal approximation, and distribution learning become chapters of the geometry of polynomials. The new contribution is negative: rounding $X \sim \mathrm{Bin}(3n,1/2)$ down to $\lfloor 2X/3 \rfloor$ destroys real-rootedness, and the generating polynomial gains a root whose imaginary part is at least $\sqrt{9n^2-9n-1}/2$, so the naive rational approximation is not strongly Rayleigh for large $n$. This matters because $\lfloor jX/k \rfloor$ rounding was proposed as a route to multivariate central limit theorems for strongly Rayleigh variables, and this paper shows that route fails already at $j=2,k=3$. What survives is that the rounded PGF is still Hurwitz stable, and small-$n$ computations indicate the best strongly Rayleigh approximation sits at distance $2/3$ in the infinity-Wasserstein metric.

What carries the argument

The load-bearing object is the probability generating function $f(u)=\prod_{i=1}^n (p_i u + 1-p_i)$, whose coefficients are the distribution's weights. The paper's organizing idea is that the coefficients form a Poisson binomial distribution if and only if $f$ is real stable, meaning all roots are real and, here, negative; this is the strongly Rayleigh property, and it is what the rounding question tests. To attack $\lfloor 2X/3 \rfloor$, the paper uses the coefficient imbalance between even and odd values, which forces a failure of Newton's inequality $a_i^2 \ge a_{i-1}a_{i+1}(1+1/i)(1+1/(n-i))$, and pairs this with root-location arguments for the lower bound on the imaginary parts. On the positive side, it uses a classical root-interlacing criterion, the Hermite-Biehler theorem, to explain why the same PGF is Hurwitz stable and hence a sum of independent random variables taking values in $\{0,1,2\}$. The approximation question is quantified by the infinity-Wasserstein distance $W_\infty$, which measures the worst-case move needed to push the rounded variable into a strongly Rayleigh distribution.

What would settle it

For $n = 10, 20, 40, 80$, compute all roots of the degree-$2n$ polynomial $\sum_k \mathbb P(\lfloor 2X/3 \rfloor = k) z^k$ with $X \sim \mathrm{Bin}(3n,1/2)$ and compare $\max_i \Im(z_i)$ with $\sqrt{9n^2-9n-1}/2$. If any $n$ has observed maximum below the bound, or growth slower than linear, Theorem 4.4 is false; matching growth would validate the missing coefficient-to-root step.

Watch

Extended reading notes

Core claim

The paper's sharpest new assertion is Theorem 4.4: for $X \sim \mathrm{Bin}(3n,1/2)$, the roots $z_i$ of the probability generating function of $\lfloor 2X/3 \rfloor$ satisfy $\max_i \Im(z_i) \ge \sqrt{9n^2-9n-1}/2$. Since the right-hand side is linear in $n$, the rounded variable is, for large $n$, far outside the strongly Rayleigh class. The explanation offered is that rounding concentrates probability unevenly on even and odd values, with $\mathbb P(\lfloor 2X/3 \rfloor=2k)$ proportional to $\binom{3n+1}{3k+1}$ and the odd masses proportional to $\binom{3n}{3k+2}$, so Newton's inequality fails. The paper also records that the same PGF is Hurwitz stable, formulates the $W_\infty$-optimal strongly Rayleigh approximation problem $\mathrm{Acc}(2X/3)$, and computes it for $n\le 6$, conjecturing that $\mathrm{Acc}(2X/3)=O(1)$.

Load-bearing premise

Theorem 4.4 hinges on a step the paper states but does not prove: that the failure of Newton's inequality for the coefficients of the generating polynomial of $\lfloor 2X/3 \rfloor$ forces some root to have imaginary part at least $\sqrt{9n^2-9n-1}/2$.

Editorial extensions

If this is right

  • For $X\sim\mathrm{Bin}(3n,1/2)$, the variable $\lfloor 2X/3\rfloor$ is not strongly Rayleigh for large $n$: its PGF has a root with imaginary part at least $\sqrt{9n^2-9n-1}/2$, so the naive integer-rounding operation leaves the real-rooted class.
  • No single binomial $\mathrm{Bin}(2n,p)$ approximates $2X/3$ in $W_\infty$ better than $C_p n$, so mean-variance matching alone cannot repair the rounding.
  • The equidistributed choice $p_i=i/(2n+1)$ gives a strictly better but still linear lower bound on $W_\infty(2X/3,\mathrm{PB}(p_1,\dots,p_{2n}))$; it is the best explicit construction the paper discusses.
  • Although $\lfloor 2X/3\rfloor$ is not strongly Rayleigh, it is Hurwitz stable, hence a sum of independent random variables taking values in $\{0,1,2\}$; the obstruction is specifically real-rootedness rather than any factorization into low-degree positive-coefficient factors.
  • The small-$n$ computations give $\mathrm{Acc}(2X/3)=1/3$ for $n=1,2$ and $2/3$ for $n=3,4,5,6$, supporting the conjecture that $\mathrm{Acc}(2X/3)=O(1)$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The coefficient-to-root mechanism behind Theorem 4.4 is generic: any fixed rounding rule $\lfloor jX/k\rfloor$ that imbalances coefficient parities should produce PGF roots with imaginary parts growing linearly in $n$, so the strongly Rayleigh class is likely closed only under exact affine maps and under $\lfloor X/k\rfloor$.
  • One can test the $O(1)$ conjecture directly by computing $\mathrm{Acc}(2X/3)$ for $n$ up to a few dozen with numerical root-finding: if the value plateaus at $2/3$ rather than decaying, the conjecture holds and the plateau value may be exactly $2/3$.
  • A quantitative lemma connecting Newton-inequality slack to root displacement would turn the paper's counterexample into a general certificate that a log-concave-but-not-PF sequence is far from every strongly Rayleigh distribution in the $W_\infty$ metric.
  • The paper's $P_3/Q_3$ examples show that root interlacing does not imply factorization into low-degree positive-coefficient polynomials, so proving Conjecture 4.8 will need something beyond the classical interlacing criterion.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper is an expository survey of the Poisson binomial distribution, covering distributional properties, Poisson/normal/binomial approximations, polynomial and strong-Rayleigh aspects, optimal transport questions around approximating 2X/3, and computational/learning results. The authors compile numerous known theorems with citations and add several new pieces: Theorem 4.4, which asserts a quantitative lower bound on the imaginary parts of the roots of the PGF of floor(2X/3) for X~Bin(3n,1/2); a discussion and open problems around the quantity Acc(2X/3); and exact values of Acc(2X/3) for small n in Appendix A. The survey portions are internally consistent and accurately attribute known results; the new mathematical claims are, however, only sketched.

Significance. If the new claims are correct, the paper would be a useful resource for the Poisson binomial community and for researchers working on strong-Rayleigh properties of discretized sums. The survey of post-2000 results on approximation, learning, and polynomial geometry is valuable and generally accurate, with citations that appear reliable. The paper also has strengths in being self-contained and in connecting diverse literatures. It does not fit free parameters to its conclusions, and its new results are checked against external benchmarks such as binomial tail estimates and Newton's inequality. The sharpest new assertion, Theorem 4.4, and the exact accuracy values in Appendix A are not yet supported by complete proofs, so the contribution as a research paper is conditional on supplying those arguments.

major comments (2)
  1. [Section 4, Theorem 4.4] Theorem 4.4 is the paper's most significant new mathematical assertion, but it is not proved. The text says only that 'one can prove' the bound and gives the observation that the coefficients of the PGF of floor(2X/3) violate Newton's inequality (4.2). That observation is qualitative: it implies the polynomial is not real-rooted, i.e. max_i Im(z_i) > 0, but it does not by itself imply the quantitative bound max_i Im(z_i) >= sqrt(9n^2 - 9n - 1)/2. The missing transfer lemma from the coefficient imbalance a_{2k} = binom(3n+1,3k+1), a_{2k+1} = binom(3n,3k+2) to a root-location bound is the entire substance of the theorem. The statement also needs a domain qualifier, since for n=1 the radicand 9n^2 - 9n - 1 is negative. Because the theorem motivates the claim that floor(2X/3) is 'far away' from being strongly Rayleigh and underpins Open Problem 4.6, this missing proof is load-bearing rather than a presentation issue.
  2. [Appendix A, n=5 and n=6] The values Acc(2X/3) = 2/3 reported for n=5 and n=6 are presented as exact, but only upper bounds are demonstrated. For n=6, the construction of Y ~ Bin(4,1/2) gives W_infinity(2X/3,Y) <= 2/3, and no argument is supplied to show that every strongly Rayleigh Y on {0,...,2n} satisfies W_infinity(2X/3,Y) >= 2/3. For n=5, the text says that a 'similar argument as in the case n=4' shows W_infinity(2X/3,Y) != 1/3, but the exclusion argument is not written out, and the possibility of values strictly between 1/3 and 2/3 is not addressed. To report these as exact values of Acc, the authors must either give the lower-bound proofs or explicitly state the values as upper bounds.
minor comments (5)
  1. [Section 3, Theorems 3.5 and 3.6] In the statements of Theorems 3.5 and 3.6, the displayed definition reads 'mu := sum_{i=1}^n p_n' where it should be 'sum_{i=1}^n p_i'; the same typo appears in Theorem 3.7.
  2. [Section 3, around (3.10)] The phrase 'Elm's approach' should be 'Ehm's approach', referring to Ehm [41].
  3. [Appendix A, n=3] The parameters written as 'PB(1/4 + sqrt(8), 1/4 - sqrt(8))' are not probabilities in [0,1]; the intended parameters appear to be approximately 1/2 + sqrt(2)/4 and 1/2 - sqrt(2)/4, matching the roots -3 ± sqrt(8) of the displayed PGF.
  4. [Appendix A, n=4] The expression 'PB(1/2 + 2/sqrt(5), 1/2 - 2/sqrt(5))' is again outside [0,1] for the plus sign; the factorized PGF (1 + 10x + 5x^2)/16 has Bernoulli parameters 5/8 + sqrt(5)/8 and 5/8 - sqrt(5)/8, so the displayed parameters should be corrected.
  5. [Section 5, (5.3)-(5.4)] The sentence 'the r.h.s of (5.3) is the discrete Fourier transform' should read 'the right-hand side of (5.3)'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the new claims are derived against external benchmarks, and the one omitted proof is a completeness gap, not a circular reduction.

full rationale

This is an expository survey whose new content, mainly in Section 4, is not fitted or defined into existence. Theorem 4.4 asserts a quantitative lower bound on the imaginary parts of roots of the PGF of floor(2X/3). The paper says only 'In fact, one can prove the following theorem' and then explains that the displayed coefficients violate Newton's inequality (4.2). That coefficient imbalance shows the polynomial is not strongly Rayleigh, but the paper does not supply the quantitative bridge from Newton-inequality failure to max_i Im(z_i) >= sqrt(9n^2-9n-1)/2. This is an omitted proof / correctness gap, not a circularity: the target quantity is not used to define the coefficients, and the assertion does not reduce to an input by construction. The other results are supported by external, non-self citations (Hoeffding, Le Cam, Barbour-Hall, Rollin, Choi-Xia, Ghosh-Liggett-Pemantle, Liggett) or by explicit small-n computations in Appendix A that check real-rootedness directly. No free parameter is fitted to a claimed prediction, no uniqueness theorem is imported from the authors' own prior work, and the authors do not make load-bearing use of their own citations. The small-n Acc values are computed by exhaustive enumeration of valid transference plans and explicit discriminant checks, so they are not renamed inputs. Overall, the derivation chain is self-contained; the only flagged issue is the unproved quantitative step in Theorem 4.4, which affects verifiability rather than circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted. The constants that appear, such as the 1/32 lower-bound constant in Theorem 3.1 or the 0.7915 in (3.7), are cited from prior work. The paper's own derivations introduce no ad hoc quantities; the W-infinity levels in Appendix A are exact rational values computed from the given distributions.

assumptions (4)
  • standard math A sequence of nonnegative numbers whose generating polynomial has only real roots is a PF sequence, and the normalized coefficients form a PB distribution (Theorem 4.1).
    Used in Section 4 to identify PB and strongly Rayleigh classes.
  • standard math Newton's inequality (4.2) is necessary and the Hutchinson-Kurtz condition (4.3) is sufficient for real-rootedness of a polynomial with nonnegative coefficients.
    Applied in Theorem 4.4 and Appendix A to test the strongly Rayleigh property.
  • standard math The binomial tail sum has asymptotic 2^(3n H(lambda)+o(n)) for lambda < 1/2 (entropy estimate).
    Used in Proposition 4.5 to derive the linear lower bound for W-infinity.
  • standard math W-infinity is a metric and optimal transference plans exist between compactly supported measures.
    Defines and justifies the optimization problem (4.6).

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Pith. "Pith review of The Poisson binomial distribution -- Old & New." pith.science (2026). https://pith.science/paper/CPNKKQIK

@misc{pith2026190810024,
  author       = {Pith},
  title        = {Pith review of: The Poisson binomial distribution -- Old & New},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CPNKKQIK}},
  note         = {Machine review of arXiv:1908.10024}
}
read the original abstract

This is an expository article on the Poisson binomial distribution. We review lesser known results and recent progress on this topic, including geometry of polynomials and distribution learning. We also provide examples to illustrate the use of the Poisson binomial machinery. Some open questions of approximating rational fractions of the Poisson binomial are presented.

Figures

Figures reproduced from arXiv: 1908.10024 by the authors.

Figure 1
Figure 1. A transference plan from 2 3 Bin(9, 1/2) to Bin(6, 1/2). In the part (1) of the program, one question is how well a Bin(2n, p) random variable for any p can approximate 2X/3. Unfortunately, the approximation is not so good as proved in the following proposition [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. (Right) shows the valid region of (θ1, θ2, θ3) [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. A transference plan from 2 3 Bin(6, 1/2) to Bin(4, 1/2). [3] T. W. Anderson and S. M. Samuels. Some inequalities among binomial and Poisson probabilities. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), pages Vol. I: Statistics, pp. 1–12. Univ. California Press, Berkeley, Calif., 1967. [4] T. Ando. Totally positive matrices. Linear Algebra Appl., 90:165–219, 1987. [5] K. A… view at source ↗

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Pith tools

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