REVIEW 2 major objections 4 minor 3 references
Limiting distributions of ratios of Binomial random variables
T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read A ratio of two independent Binomial counts, after centering and scaling, converges in distribution to a Normal variable with an explicit variance, in each of three regimes for the trial counts.
desk verdict Intended result is sound and worth refereeing, but Theorem 1 as printed swaps r and s, which is a statement-level typo that must be fixed. 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 load-bearing object is the second-order Taylor expansion of $f(x,y)=x^s/(x+y)^r$ at the mean point $(np,mp)$, which decomposes $R$ into a deterministic center, a Normal-looking linear term, and a quadratic remainder $Q(X,Y)$. The remainder is controlled on the high-probability event $A$ where $|X-np|\le C\sqrt{n\log n}$ and $|Y-mp|\le C\sqrt{m\log m}$; on $A$, the spectral norm of the Hessian $\nabla^2 f$ is bounded using a standard eigenvalue-localization argument, yielding $|Q(X,Y)|\le C n^{s-2}\log(n+m)/(n+m)^{r-1}$. Lemma 1 of the paper, a standard result that a sequence converging in distribution plus a term converging to zero in probability has the same limit, then allows the scaled remainder to be dropped. The three regimes in Theorem 1 are exactly the choices of $m/n$ for which the scaled remainder bound vanishes.
What would settle it
Simulate $r=s=1$, $p=0.5$, $n=m=10^6$, and compare the empirical variance of $\sqrt{n}(X/(X+Y)-1/2)$ with the theorem's value $(1-p)/(8p)=0.125$; a material mismatch would refute case (ii). A second, sharper check: take $s=0.5$, $r=2$ with $m=\alpha n$ for several $\alpha$ and verify the case-(ii) variance formula, since the remainder bound is least secure for small $s$ near $x=0$.
Extended reading notes
Core claim
The central claim is that, in the notation of the abstract and proof, $R=X^s/(X+Y)^r$ converges in distribution to a zero-mean Normal variable after centering and scaling, in three regimes: (i) $m/n\to\infty$ with $m\log m\,n^{-3/2}\to 0$, with scaling $m^r/n^{s-1/2}$ and variance $p^{2(s-r)-1}(1-p)s^2$; (ii) $m/n\to\alpha\in(0,\infty)$, with scaling $n^{r-s+1/2}$ and variance $p^{2(s-r)-1}(1-p)[(s(1+\alpha)-r)^2+\alpha r^2]/(1+\alpha)^{2(r+1)}$; and (iii) $m/n\to 0$, with scaling $n^{r-s+1/2}$ and variance $p^{2(s-r)-1}(1-p)(s-r)^2$. The proof shows that after the appropriate scaling the quadratic remainder of a Taylor expansion vanishes in probability, so the limiting distribution is carried entirely by the linear term, a linear combination of two independent standardized Binomial variables.
Load-bearing premise
The proof rests on the bound that the quadratic remainder of the Taylor expansion is small on the very likely event that both binomial counts stay within about $\sqrt{n\log n}$ and $\sqrt{m\log m}$ of their means; if that bound fails or is not uniform for all $r,s>0$, the Normal convergence could fail.
Editorial extensions
If this is right
- In the regime where the denominator count grows much faster than the numerator count, the theorem gives a usable Normal approximation with variance $p^{2(s-r)-1}(1-p)s^2$, governed by the numerator's parameter $s$ alone.
- When $m/n\to\alpha$, the variance formula interpolates between the two extremes and shows how the relative size of the two Binomial trials shapes the fluctuation of the ratio.
- In the regime $m/n\to 0$, the limiting variance is proportional to $(s-r)^2$, so the scaled ratio collapses to a point mass when $r=s$; the simulations show a corresponding spike in KL divergence at that degeneracy.
- The paper notes that the previously studied case $m\sim n^2$ with $r=1,\ s=1/2$ is not covered by these regimes, and that higher moments would require a sharper treatment of the quadratic remainder.
Reading between the lines
- Beyond the paper: because the remainder bound is explicit in $n$ and $m$, the same argument should yield a convergence rate (Berry-Esseen-type) for the Normal approximation, which would tell practitioners how large the trial counts must be.
- Beyond the paper: the boundary case $m/n\to\infty$ with $m\log m\,n^{-3/2}\not\to 0$ is outside the theorem; the paper's own simulations show the distribution collapsing toward a point mass there, suggesting that a different, possibly non-Gaussian limit governs that intermediate regime.
- Beyond the paper: the proof only uses independence, the mean-variance structure, and sub-Gaussian tail bounds of $X$ and $Y$, so the theorem plausibly transfers to independent sums of bounded, zero-mean, identically scaled increments, with $p(1-p)$ replaced by the common variance of the increments.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper establishes a central limit theorem for the ratio R = X^s/(X+Y)^r, where X ~ Binomial(n,p) and Y ~ Binomial(m,p) are independent and r,s > 0 are fixed. Under three asymptotic regimes for m/n — namely m/n → ∞ with m log(m)/n^{3/2} → 0, m/n → α ∈ (0,∞), and m/n → 0 — the authors show that, after centering by n^s/(n+m)^r p^{s-r} and scaling by either m^r/n^{s-1/2} or n^{r-s+1/2}, the ratio converges in distribution to a zero-mean Normal law with an explicit variance. The proof is based on a second-order Taylor expansion of f(x,y) = x^s/(x+y)^r, a Gerschgorin-type bound on the Hessian, concentration inequalities for binomial random variables, and an application of Slutsky's lemma. The theoretical results are illustrated with simulations that compare the empirical distribution of the scaled ratio to the predicted Normal using KL divergence.
Significance. If the intended result is correct, the paper provides a clean and explicit asymptotic normality theorem for a class of ratios of independent binomials, extending the special case r=1, s=1/2 treated in the earlier work of Lubberts et al. The derivation is transparent and parameter-free: the mean and variance are obtained analytically from the Taylor expansion with no constants fit to simulations. The three regimes cover a natural range of relative growth rates of the two binomial sample sizes, and the simulations provide supporting evidence. The paper is short and readable, and the main mathematical claim appears sound.
major comments (2)
- [Theorem 1] The statement of Theorem 1 defines R = X^r/(X+Y)^s, while the abstract, the introduction, Eq. (1), the centering term n^s/(n+m)^r p^{s-r}, the variance formulas, and the simulations all use R = X^s/(X+Y)^r. With the printed definition, the theorem is false as stated because the centering exponent and the variances correspond to the other ratio. Please correct the definition of R in Theorem 1 to match the rest of the manuscript.
- [Section 2, Eq. (3)] The derivation of the remainder bound (3) is too terse to be fully checkable. In particular, the step 'the right hand side of inequality (2) may be bounded by ...' needs a detailed verification that, on the event A, the sum of the absolute entries of the Hessian is uniformly bounded by C x^{s-2} max{s(n+m), rn}^2/(x+y)^{r+2} for all r,s > 0. This includes the delicate case s < 2, where x^{s-2} is unbounded and the bound relies on x staying bounded away from 0 on A; please spell out the required inequalities and the choice of the constant C.
minor comments (4)
- [Section 2, event A] The event A is first introduced as |X-np| ≤ sqrt(n log n) p(1-p), without the constant C; the subsequent paragraph and the bound (3) use C sqrt(n log n). Please make the definition consistent and include the constant from the outset.
- [Section 2, proof of Theorem 1] The convergence of the scaled linear term to the specified Normal distribution is stated informally ('the last vector will converge to a Normal vector') but not written out. A short explicit computation of the limiting variance from the coefficients would improve readability.
- [Section 3, Eq. (4)] The KL divergence formula (4) may involve log(0) if a bin has zero observed proportion in one of the samples; please state how empty bins are handled in the implementation.
- [Section 2, remainder estimate] The notation r_x and r_y for the deviations of X and Y from their means is introduced but the subscript is nonstandard; consider using δ_x and δ_y to avoid confusion with the parameter r.
Circularity Check
No circularity; the limiting-distribution claim follows from an explicit Taylor expansion with a bounded remainder, and the only self-citation is motivational and not load-bearing.
full rationale
The derivation of Theorem 1 is self-contained: the paper expands R=X^s/(X+Y)^r about (np, mp), identifies the linear term as the asymptotic Normal part, and bounds the quadratic remainder via the Hessian bound (2) and high-probability event A, yielding bound (3). The three regime scalings are then checked against (3) to make the remainder vanish. No parameter is fitted from the simulations; the means and variances come from the Taylor coefficients, and the simulations only illustrate the result. The only citation to the authors' prior work is [3], where the motivating case r=1, s=1/2 appeared; that result is not used in the proof here. One non-circular defect exists: Theorem 1 defines R=X^r/(X+Y)^s, while the abstract, introduction, Eq. (1), centering constants, variances, and simulations all use R=X^s/(X+Y)^r. Taken literally the printed statement's centering would be wrong, but this is a typo in the definition, not an argument that reduces to its inputs. Thus there is no significant circularity.
Assumptions & free parameters
assumptions (5)
- standard math Taylor's theorem with second-order remainder for f(x,y)=x^s/(x+y)^r (equation 1).
- standard math Slutsky-type lemma (Lemma 1): if X_n converges in distribution to X and Y_n converges to 0 in probability, then X_n+Y_n converges in distribution to X.
- standard math Gerschgorin disk theorem to bound the spectral norm of the Hessian.
- domain assumption Binomial concentration: |X-np| ≤ C sqrt(n log n) p(1-p) with probability at least 1 - C n^{-2}, and similarly for Y.
- domain assumption X and Y are independent Binomial(n,p) and Binomial(m,p) with common p∈(0,1).
Cite this review
Pith. "Pith review of Limiting distributions of ratios of Binomial random variables." pith.science (2026). https://pith.science/paper/I3TDVFED
@misc{pith2026250613071,
author = {Pith},
title = {Pith review of: Limiting distributions of ratios of Binomial random variables},
year = {2026},
howpublished = {\url{https://pith.science/paper/I3TDVFED}},
note = {Machine review of arXiv:2506.13071}
}
abstract
We consider the limiting distribution of the quantity $X^s/(X+Y)^r$, where $X$ and $Y$ are two independent Binomial random variables with a common success probability and a number of trials $n$ and $m$, respectively, and $r,s$ are positive real numbers. Under several settings, we prove that this converges to a Normal distribution with a given mean and variance, and demonstrate these theoretical results through simulations.
Figures
Figures from the paper (1 more)
Reference graph
Works this paper leans on
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[3]
Random line graphs and edge-attributed network inference
Zachary Lubberts, Avanti Athreya, Youngser Park, and Carey E Priebe. Random line graphs and edge-attributed network inference.Bernoulli, 2025. arXiv:2103.14726. 9
work page Pith review arXiv 2025
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[1]
Elsevier, 2000
Kai Lai Chung.A course in probability theory. Elsevier, 2000
2000
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[2]
Cambridge university press, 2012
Roger A Horn and Charles R Johnson.Matrix analysis. Cambridge university press, 2012
2012
Reviewed August 7, 2026 · model on record in the stance chip above.
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