REVIEW 2 major objections 3 minor 1 cited by
Binomial probabilities at a fixed distance from the mode: size-biasing and the complete asymptotic expansion
T0 review · 2 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash
Pith's one-line read This paper proves that binomial probabilities at a fixed integer distance from the mode have a complete asymptotic expansion in powers of 1/N, uniform in p and in the shift, with the oscillating fractional part of the mean encoded in Bernou
desk verdict Solid structural companion that turns a known all-order expansion into clean Appell form, but the load-bearing base expansion is imported from an unrefereed companion preprint. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The size-bias identity m b(m;N,p) = Np b(m−1;N−1,p) is the load-bearing identity: it converts the logarithmic tail of the naive expansion into the exact prefactor Np/m. The two Appell sequences — the Bernoulli polynomials B_n(t) for the size-biased normalisation and the even sequence bB_n(t) = (B_n(t)+B_n(t+1))/2, generated by (z/2)coth(z/2)e^{tz}, for the symmetric normalisation — carry the coefficients. The difference identity B_{k+1}(h+r)−B_{k+1}(h) = (k+1) Σ_{j=0}^{r-1}(h+j)^k is what reduces the shift r to finite power sums.
What would settle it
Compute the exact probability Pr{X = ⌈Np⌉+r} for p = 1/√7 and r = 3 for N = 10^4, 10^5, 10^6 using the gamma-function representation of the binomial coefficient, and compare with Theorem 5.1 truncated after M = 2 terms. The relative error should decrease by factor ~10^3 for each factor of 10 in N, uniformly in the oscillating h_N; any slower decay refutes the uniformity claim. Alternatively, verify the exact identity (28) at a non-integer h (say h=1/3, r=2, p=1/3): both sides are rational polynomials in h and p, so the equality can be checked symbolically; a single mismatch refutes the Bernoul
Extended reading notes
Core claim
The paper's central claim is Theorem 5.1: as N→∞, uniformly for p in a compact subset of (0,1) and integers r with |r|≤R, Pr{X=⌈Np⌉+r} ∼ (Np/(Np+h+r)) (2πNpq)^{-1/2} exp( Σ_{k≥1} [eA_k(h;p) − Ξ_k(p) S_k(h,r)/k] / N^k ), where h = ⌈Np⌉ − Np, eA_k are built from Bernoulli numbers and Bernoulli polynomials, Ξ_k(p) = (−1)^{k+1}p^{-k}+q^{-k}, and S_k(h,r) is a finite power sum along h, h+1, ..., h+r−1. Equivalently, after a symmetric normalisation the prefactor becomes the classical Stirling factor and the power sum becomes trapezoidal. The paper shows that the elementary tail of the previously known expansion is exactly the size-bias factor, so the 'pure' coefficients are Appell; that the intege
Load-bearing premise
The entire expansion rests on the un-reproved companion theorem [7, Theorem 3.1] asserting a complete uniform asymptotic expansion of the binomial local mass at arbitrary real displacement t, with relative remainder O(N^{-M-1}) on compacta; if that quoted expansion fails in uniformity, the present Theorem 5.1 loses its error term as well.
Editorial extensions
If this is right
- Every fixed-distance local mass of a binomial law is now known to all orders in 1/N, with the fractional part of the mean preserved in the coefficients; this is directly useful for likelihood approximations and continuity corrections near the mode.
- The exact mode rule — upper mode at ⌈Np⌉ when h≤p, at ⌈Np⌉−1 when h>p — is reproduced by the first asymptotic coefficient, so the expansion correctly encodes the lattice rounding, not just the smooth skewness correction.
- The central-binomial expansions of the literature (p=1/2, N even) are recovered as the special case h=0, and the vanishing of even-order coefficients at p=q explains the alternating structure there.
- The mean absolute deviation of the binomial, via De Moivre's identity, inherits a pure Appell expansion, resolving the term-by-term cancellation observed in the companion paper.
- Averaged over N, the oscillating coefficients have closed-form Cesàro limits: zero for irrational p (by equidistribution) and the explicit periodic mean b^{-j}B_j for rational p=a/b, which gives a smooth all-purpose approximation to the modal expansion.
Reading between the lines
- The same size-bias-plus-Appell mechanism should transfer to other lattice distributions possessing a size-bias identity — Poisson, negative binomial, hypergeometric — replacing the Bernoulli sequence by the appropriate Appell generator; the paper suggests this but does not carry it out.
- In the crossover region r ≍ √N, where the fixed-distance expansion loses its ordering, one could interpolate between the Bernoulli-polynomial coefficients and the Hermite-polynomial Edgeworth expansion; the paper identifies the regime but leaves the transition unexplored.
- The exact Cesàro-average coefficients (Proposition 8.1) imply a practical smoothing: for a single large N and unknown fractional part, one can use the averaged coefficients, with the error controlled by the discrepancy of the sequence {Np}; this is a testable approximation that the paper does not formulate.
- Because Corollary 5.5 gives the modal window sum in closed form as a cubic in R, it offers a fast way to compute local coverage probabilities without summing the individual probabilities; this may be useful in confidence-interval calculations, though the paper does not pursue that application.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper considers X ~ Bin(N,p), nu = ceil(Np), h_N = nu - Np, and gives asymptotic expansions for Pr{X = nu + r} as N tends to infinity, uniformly for p in compact subsets of (0,1) and |r| <= R. The starting point is a quoted theorem ([7, Thm 3.1], stated as Theorem 2.1) providing a complete local expansion with coefficients A_k(t;p). The paper's contribution is structural: it identifies the non-Bernoulli tail in A_k with log(m/(Np)) and hence with the binomial size-bias factor; removes it to obtain a 'pure Appell' normalization; introduces a second, p-to-q symmetric normalization using the even Appell sequence hat B_n; proves that the integer shift r enters only through finite power sums S_k or T_k (the Euler-Maclaurin correspondence); and assembles these into Theorem 5.1. It also derives the exact mode rule from the first-order coefficient, verifies the expansions by exact ratio identities and by numerical checks to about 60 digits, and computes Cesaro averages of the oscillating coefficients.
Significance. If the base expansion is valid, the paper delivers a genuinely complete, uniformly valid expansion for fixed-distance modal binomial probabilities in the regime where the fractional part h_N oscillates. The identification of the tail with the size-bias factor is elegant and explains the earlier cancellation in the companion paper on mean absolute deviation. The Euler-Maclaurin split shows that the r-dependence is elementary, and the two normalizations clarify the trade-off between Appell purity and p-to-q symmetry. The paper is transparent about its dependence on [7]; the internal algebra is coherent, and the claimed agreement with special cases and with exact ratio identities is reassuring. These structural results are likely to be useful for lattice local limit problems and for the associated mean-deviation literature.
major comments (2)
- [Section 2, Theorem 2.1; Remark 5.2; Theorem 5.1] The main expansion is not self-contained. Theorem 5.1 (Eqs. (29)-(30)) inherits its existence, uniformity in p and t, and the O(N^{-M-1}) relative remainder entirely from the quoted companion theorem [7, Thm 3.1], stated here as Theorem 2.1. The paper explicitly says in Remark 5.2 that Theorem 2.1 is quoted, not reproved. This is load-bearing: if [7, Thm 3.1] is not available, has a hidden hypothesis, or fails to be uniform in t, then Theorems 2.4, 3.3 and 5.1 collapse. The numerical checks in Section 9 verify specific rates and identities but do not establish the uniform remainder. I request that a proof of Theorem 2.1 be included in an appendix or provided through a published reference; without this, the central claim remains conditional on an external same-author preprint.
- [Section 7.1, Eq. (34)] The sentence 'Theorem 5.1 must reproduce this identically' overstates the logical status. Theorem 5.1 is an asymptotic statement with a formal infinite exponent, not a theorem that yields exact finite-N identities. The displayed derivation shows that the formal series for the exponent difference telescopes to the logarithm of the exact ratio, which is a useful algebraic consistency check; however it is not a consequence of the remainder theorem. Please rephrase (and perhaps note that the matching is exact only in the formal or convergent-series sense) to avoid implying that an asymptotic theorem proves an exact identity.
minor comments (3)
- [Section 7.2] The claim that [3, Theorem 3.1] contains a sign misprint in P_1 would be easier to check if the relevant formula from [3] were reproduced alongside the present derivation.
- [Section 8, Proposition 8.1] The proof of equidistribution of h_N for irrational p is compressed; since h_N is 1 - {Np} except at atoms, a one-line reduction to Weyl's theorem would be clearer.
- [Section 9] The numerical checks are extensive, but the paper does not include code or exact evaluation details. A short reproducibility note would be useful, though this is not required for the mathematics.
Circularity Check
Central expansion rests on a same-author companion theorem that is quoted, not reproved; the structural identifications are independent.
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self citation load bearing
[Section 2 (Theorem 2.1) and Remark 5.2]
"We take as given the following, which is [7, Theorem 3.1]. ... The existence of an expansion at t=h+r follows at once from [7, Theorem 3.1] by substituting the bounded displacement t=h+r — this is the role of the uniformity in t, as noted in §2. ... Theorem 2.1 is quoted, not reproved."
Theorems 2.4, 3.3, and 5.1 all inherit their existence, uniformity in p and t, and the relative remainder O(N^{-M-1}) from [7, Theorem 3.1], a companion paper by the same author that is not proved here. Thus the main theorem's convergence and remainder are not self-contained; if the quoted theorem or its uniformity fails for the oscillating t=h+r, the central expansion collapses. This is load-bearing self-citation rather than a definitional circle, because the Appell/size-bias splitting and the Euler-Maclaurin r-dependence are proved from exact identities.
full rationale
No fitted-input circularity, no Ansatz smuggling, and no renaming of a known result were found. The size-bias factorization (9)/(14), the coefficient splittings (28), and the Appell-sequence identities are exact algebraic manipulations proved in the paper; the mode rule and central-binomial checks are consistency checks, not fits. The only circularity concern is the paper's explicit reliance on the author's own [7, Theorem 3.1] as the unproved base expansion. Because that theorem is quoted rather than reproved and is load-bearing for the remainder estimate of the main theorem, the paper cannot be scored 0; however, the central structural claims have independent content and the dependency is openly stated, so a moderate score of 4 is appropriate rather than a charge of full circularity.
Assumptions & free parameters
assumptions (4)
- domain assumption Theorem 2.1 ([7, Thm 3.1]): π_N(t;p) ∼ (2πNpq)^{-1/2} exp(Σ A_k(t;p)/N^k) uniformly in p∈K, t∈H, with relative remainder O(N^{-M-1}).
- standard math Bernoulli Appell identities: B_n(t+1)-B_n(t)=n t^{n-1}; B_n(1-t)=(-1)^n B_n(t); binomial translation formula.
- standard math Weyl equidistribution of ({Np}) for irrational p; Bernoulli multiplication theorem Σ_{i=0}^{b-1} B_j(i/b)=b^{1-j} B_j(0).
- domain assumption Admissibility: Np+t ∈ {0,...,N} and p bounded away from 0 and 1, t bounded; N large enough so m and N−m are order N.
Cite this review
Pith. "Pith review of Binomial probabilities at a fixed distance from the mode: size-biasing and the complete asymptotic expansion." pith.science (2026). https://pith.science/paper/MPF3IP6F
@misc{pith2026260719844,
author = {Pith},
title = {Pith review of: Binomial probabilities at a fixed distance from the mode: size-biasing and the complete asymptotic expansion},
year = {2026},
howpublished = {\url{https://pith.science/paper/MPF3IP6F}},
note = {Machine review of arXiv:2607.19844}
}
abstract
We study binomial probabilities at a fixed integer distance from the upper mode, in the regime where the fractional part of the mean remains visible. The complete asymptotic expansion is obtained to all orders, uniformly for the success probability in compact subintervals of $(0,1)$ and for bounded shifts from the mode. The main point is structural. A logarithmic tail in the standard local expansion is exactly the binomial size-bias factor, and removing it leaves a pure Appell expansion. A second, symmetric normalisation is governed by an even Appell sequence and recovers the classical Stirling prefactor. The resulting coefficients explain the exact binomial mode rule, recover the central-binomial expansions as a special case, and give closed forms for averaged oscillating coefficients.
Forward citations
Cited by 1 Pith paper
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The mean absolute deviation of the classical discrete distributions: collapse identities, complete asymptotic expansions, and enveloping series
A common telescoping identity yields closed forms, complete asymptotics, and enveloping series for the mean absolute deviation of the four classical discrete laws.
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