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Local binomial expansions with an Appell shift, and the mean absolute deviation of the binomial distribution

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Pith's one-line read For any binomial distribution, the mean absolute deviation of X from its mean has a complete asymptotic expansion in powers of 1/N, with coefficients that are explicit Bernoulli polynomials of the fractional displacement of the mean.

desk verdict Genuinely new coefficient structure for binomial MAD, with a uniformity gap in Lemma 2.1 that needs a referee's attention; otherwise a solid, citable paper. read the letter →

arxiv 2607.18494 v1 pith:KL6N7SHN submitted 2026-07-20 math.CA math.PR

Neven Elezović This is my paper · ORCID
classification math.CAmath.PR MSC 41A6011B6860C0533B15
keywords binomialdistributionmeanabsolutedeviationasymptoticexpansionBernoullipolynomialsAppellgammaquotientlatticecorrectionStirlingseries
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 establishes that for X~Bin(N,p), the quantity E|X−Np| has a full asymptotic expansion in powers of N−1 relative to the Gaussian leading term √(2Npq/π), with coefficients that depend on the bounded, non-convergent displacement h_N = ⌈Np⌉−Np through Bernoulli polynomials. The route is exact: De Moivre's identity collapses the expectation to a single binomial mass, which is then written as a ratio of three gamma functions with unequal linear scalings. The central structural cancellation—the elementary part of the shift cancels term by term against the prefactor's logarithm—leaves coefficients that are pure Bernoulli polynomials, so the expansion is both complete and algebraically transparent. This fills a gap that stopped after the first correction, and it supplies rigorous two-sided bounds as well as an enveloping sign-alternating series when the mean is an integer.

What carries the argument

The central object is a gamma-quotient expansion for ratios of the form Γ(N+1)/(Γ(Np+h+1)Γ(Nq−h+1)) with arbitrary real scalings p,q, stated with uniformity in a compact set of shifts. It extends the usual Stirling series—where all arguments share the same scale—by weighting each contributing Bernoulli polynomial B_{n+1}(u) by the (−n)-th power of that factor's own scaling λ_j. This expansion is combined with two structural identities of Appell sequences: the difference identity B_m(h+1)=B_m(h)+m h^{m−1} and the reflection identity B_m(1−h)=(−1)^m B_m(h), which reduce the three gamma shifts to a single Bernoulli polynomial B_{k+1}(h) plus an elementary residue. The term-by-term cancellation

What would settle it

Fix a truncation order M and a compact subinterval K of (0,1). Take a sequence of N with h_N tending to 1 and p tending to the left endpoint of K, and compute the exact remainder R_M(N,p) = log(E|X−Np|/√(2Npq/π)) − Σ_{k=1}^M a_k(h_N;p)/N^k. If |R_M(N,p)|·N^{M+1} is unbounded over such a sequence, the uniform statement (26) fails. Separately, for N p integer, checking a single violation of the strict envelope inequality (38) would refute the enveloping claim.

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Extended reading notes

Core claim

The paper's main theorem states that, as N→∞, uniformly for p in a compact subinterval of (0,1), E|X−Np| = √(2Npq/π) exp(Σ_{k≥1} a_k(h_N;p)/N^k), with a_k(h;p) = [(-1)^{k+1}/(k(k+1))] [ B_{k+1} − (p^{-k} + (-1)^{k+1} q^{-k}) B_{k+1}(h) ]. Here B_{k+1} are the Bernoulli polynomials, h_N = ⌈Np⌉−Np is the bounded, oscillating displacement of the mean above an integer, and all powers of N are integer relative to the leading factor—no half-integer corrections occur at any depth. The derivation reduces the expectation, via the exact identity, to the binomial mass at the first lattice point above the mean, writes that mass as a gamma quotient, and expands it using a version of the Stirling series w

Load-bearing premise

The load-bearing premise is that the gamma-quotient expansion is uniform in the bounded displacement h and in the scalings p, q bounded away from zero; if the remainder cannot be uniformly controlled as p approaches 0 or 1, then the expansion is empty because the displacement h_N does not converge.

Editorial extensions

If this is right

  • The classical first-order correction to the Gaussian mean deviation, known since the 1940s, is recovered as the coefficient a_1(h_N;p) of the new all-order expansion; all higher-order corrections are now generated by a single recursion.
  • Because the coefficients carry the oscillating factor h_N, the expansion explains the non-uniformity of the Gaussian approximation near p=0 or 1: coefficients grow like (pq)^{-m} as the endpoints are approached.
  • At an integer mean (Np ∈ Z) the logarithmic expansion collapses to a sign-alternating series in odd powers of N^{-1}; the paper proves that this series is enveloping, so successive truncations bracket the true expectation, giving rigorous two-sided inequalities.
  • A universally valid two-sided bound of logarithmic width O(N^{-2}) is obtained by combining the exact reduction with Robbins's sharpened Stirling inequality, holding uniformly in the lattice position for all interior p.
  • The Cesàro means of the oscillating coefficients are explicit: for irrational p the mean is the smooth part δ_m(p), and for rational p=a/b the mean over a period is obtained by replacing B_j(h) with b^{-j}B_j, a direct consequence of the Bernoulli multiplication theorem.

Reading between the lines

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

  • The same gamma-quotient machinery with unequal scalings is likely to extend to the multivariate analogue—the expected total-variation-type statistic Σ_j |X_j − Np_j| for multinomial variables—where no closed form exists; the single-bin expansion may serve as the exactly solvable template for the general lattice Edgeworth expansion.
  • The cancellation between the prefactor's logarithm and the difference-residue of the Bernoulli shift suggests a general principle for Appell sequences: whenever a prefactor coincides with the argument of a shifted gamma quotient, the non-polynomial residue cancels, leaving pure Appell polynomials; this could be tested on other Appell systems such as Euler polynomials.
  • The enveloping property is proven only for integer means; for non-integer means the sign of the remainder after two terms is expected to oscillate with h_N, but a stochastic or averaged version might hold—examining the Cesàro means of the truncation errors would be a concrete testable extension.
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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

1 major / 3 minor

Summary. The paper derives complete asymptotic expansions, as N→∞, for the binomial local mass at a bounded lattice displacement and for the mean absolute deviation of X∼Bin(N,p). Starting from De Moivre's exact formula, the MAD is reduced to a single local mass at ν=⌈Np⌉. The authors develop a gamma-quotient expansion with unequal scalings (Lemma 2.1) and show, in Theorems 3.1 and 4.2, that the coefficients are built from Bernoulli polynomials evaluated at the oscillating displacement h_N=⌈Np⌉−Np; the elementary part cancels term by term. At integer mean, the expansion collapses to a sign-alternating odd series, and Theorem 5.5 proves that this series is enveloping via a sign-definite Binet-kernel representation. The paper also gives an elementary two-sided bound (Proposition 5.1) and Cesàro means of the oscillating coefficients (Proposition 6.1).

Significance. If the main results are correct, the paper solves a classical problem in full generality: it extends the first-order Frame–Johnson correction to all orders and identifies the Appell-shift mechanism. The exact cancellation, the enveloping property at integer means, and the simple proof of the two-sided bound are attractive. The paper is self-contained and uses only classical tools, and it gives a nice illustration of Bernoulli polynomials in a probabilistic context. The main risk is the uniformity in Lemma 2.1, on which the central Theorems 3.1 and 4.2 rely.

major comments (1)
  1. [§2, Lemma 2.1, Eq. (11)] The uniformity statement (11) is load-bearing for Theorems 3.1 and 4.2, but its proof is only sketched. After truncating the unshifted Stirling series, the re-expansion of (1+u_j/(λ_jx))^{1-2n} produces contributions at all orders in x^{-1}. The text asserts that 'collecting the powers' yields the truncated shifted expansion with a uniform O(x^{-M-1}) remainder, but it does not show that each coefficient of x^{-k}, k≤M, involves only finitely many n, nor does it track the dependence of the remainder on H and c. Please expand this argument or give a precise reference for the uniform unequal-scaling version. This is necessary because h_N does not converge and p ranges over a compact set.
minor comments (3)
  1. [§6, heading] The section heading 'A veraging the oscillation' appears to be a typo; it should be 'Averaging the oscillation'.
  2. [§5, Theorem 5.5, Eq. (39)] The exact identity (39) is stated without derivation. Adding the algebra showing that the +1 shifts in the gamma arguments assemble into the leading factor would improve readability.
  3. [§2, proof of Lemma 2.1] In the phrase 'the remainder is O(x^{-M-1}) uniformly', it would be useful to restate explicitly that the implied constant depends on M, H, c, and C.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained, starting from De Moivre's exact identity and standard Stirling expansions; Bernoulli polynomials enter through algebraic shift identities, and the key cancellation in Theorem 4.2 is derived, not assumed.

full rationale

The chain of derivation is not circular. The mean absolute deviation is first reduced exactly by De Moivre's formula (2) to a single binomial mass, and the mass is written as an exact gamma quotient. The expansion of that quotient (Theorem 3.1) is obtained by applying the gamma-quotient lemma (Lemma 2.1), whose proof is given in Section 2 and rests on classical Stirling remainder estimates and elementary Taylor re-expansion; it does not assume the target result. The Bernoulli-polynomial form of the coefficients is produced by applying the shifted Stirling expansion and using standard identities such as B_{k+1}(h+1)=B_{k+1}(h)+(k+1)h^k and the reflection identity for Bernoulli polynomials. No parameter is fitted to data, and no 'prediction' is a renamed input. In Theorem 4.2, the cancellation between the logarithmic prefactor log(1+h_N/(Np)) from (19)-(20) and the elementary residue (-1)^k h^k/(k p^k) in A_k of (17) is an explicit, term-by-term exact algebraic identity demonstrated in the proof; it is not built into a definition. The Frame-Johnson first correction is invoked as an independent check, the integer-mean expansions are verified against the classical central binomial coefficient, and the numerical table compares against exact values computed from (2), which is validation rather than fitting. The citations to the author's earlier work, especially [10] for the Appell-shift viewpoint, are not load-bearing: the uniform shifted Stirling formula is also referenced to Luke [22] and can be verified directly from the standard unshifted Stirling expansion. The uniformity assertion in Lemma 2.1 is somewhat compressed and could be a correctness/rigor concern, but that is a verification weakness, not circularity, since it does not assume the conclusion being derived. Overall the central claims have independent content and are not forced by definition or by a self-citation chain.

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

No free parameters are fitted; p, q, and h are the problem's inputs. The paper relies on standard gamma-function analysis (Stirling, Binet, Mittag-Leffler) and a classical equidistribution theorem, all properly attributed.

assumptions (4)
  • standard math Standard Appell form of Stirling's expansion: log Γ(x+t) ~ ... with Bernoulli polynomials B_{n+1}(t); uniform for t in compact sets.
    Used throughout as the basis of Lemma 2.1; cited from Luke [22] and the author's own [10].
  • standard math Robbins's sharpened Stirling bounds (31): sqrt(2πn)(n/e)^n e^{1/(12n+1)} < n! < sqrt(2πn)(n/e)^n e^{1/(12n)}.
    Used in Proposition 5.1 for the two-sided bound.
  • standard math Binet's first formula for log Γ and the Mittag-Leffler expansion of the hyperbolic cotangent kernel.
    Used in Theorem 5.5 to get the sign-definite integral representation (39)-(40).
  • standard math Weyl equidistribution theorem for irrational multiples.
    Used in Proposition 6.1(i) to obtain Cesàro means.

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Pith. "Pith review of Local binomial expansions with an Appell shift, and the mean absolute deviation of the binomial distribution." pith.science (2026). https://pith.science/paper/KL6N7SHN

@misc{pith2026260718494,
  author       = {Pith},
  title        = {Pith review of: Local binomial expansions with an Appell shift, and the mean absolute deviation of the binomial distribution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KL6N7SHN}},
  note         = {Machine review of arXiv:2607.18494}
}
abstract

We derive complete asymptotic expansions for the binomial mass at a bounded lattice displacement and for the mean absolute deviation $E|X-Np|$, $X\sim Bin(N,p)$, with $0<p<1$. De Moivre's exact formula reduces the latter problem to the local mass at $\nu=\lceil Np\rceil$, so the coefficients depend on the oscillating displacement $h_N=\lceil Np\rceil-Np$. We show that the full expansion is governed by Bernoulli polynomials evaluated at this displacement; equivalently, the lattice correction is an Appell shift in the Stirling series. The calculation is based on a gamma-quotient expansion with unequal linear scalings, stated with uniformity in the bounded shift. In passing from the local mass to the mean absolute deviation, the elementary, non-Bernoulli part of the shift cancels term by term against the De Moivre prefactor, leaving coefficients that are pure Bernoulli polynomials. As consequences, the classical first correction of Frame and Johnson is embedded in the general coefficient sequence, and the Ces\`aro means of the oscillating coefficients are obtained from the multiplication theorem for Bernoulli polynomials. Finally, although an oscillating asymptotic expansion does not bound its function by truncation, De Moivre's identity together with Robbins's form of Stirling's formula yields an elementary two-sided bound of logarithmic width $O(N^{-2})$ in the interior; and at an integer mean the logarithmic expansion reduces to a sign-alternating series in odd powers of $N^{-1}$ which we prove to be enveloping, via a sign-definite Binet-kernel representation of the combined three-gamma Stirling remainder: successive truncations bracket the mean absolute deviation.

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Forward citations

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