{"id":"877ee237-c11d-42da-afe5-43a2f1282968","arxiv_id":"2506.13071","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For independent Binomials with common p, the centered and scaled ratio X^s/(X+Y)^r converges in distribution to a Normal law in three regimes of m/n.","lead":"This paper derives the approximate bell-curve distribution of ratios X^s/(X+Y)^r when X and Y are independent Binomial counts with the same success probability and large numbers of trials. The result gives explicit mean and variance formulas, with simulations confirming the fit in several growth regimes for the trial counts.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant mathematical objection identified to the intended CLT; Theorem 1's exponent swap is a statement-level typo, not a flaw in the argument.","rationale":"I re-derived the linear term: with R=X^s/(X+Y)^r, ∇f at (np,mp) gives coefficients proportional to s(n+m)/n - r for X and -r for Y. After the theorem's scaling, case (m/n→∞) leaves only the X coefficient s, giving variance s^2 p^{2(s-r)-1}(1-p); case (m/n→α) gives the stated numerator (s(1+α)-r)^2 + αr^2 over (1+α)^{2(r+1)}; and case (m/n→0) leaves (s-r)^2. The remainder term is bounded by Cauchy-Schwarz on the event A; the Hessian entries are O(n^{s-2}/(n+m)^r) and the squared deviation is O((n+m)log(n+m)), so |Q| is bounded by C n^{s-2} log(n+m)/(n+m)^{r-1}. The paper's scaling computations in Theorem 1's proof correctly show this bound vanishes in each regime. I did not find a hidden failure for r,s>0: event A keeps the line segment away from the singularity at x=0, and the constants C depend only on fixed r,s,p. The reader's weakest assumption (the bound (3)) is plausible and I independently confirmed it to leading order. The real issue is the mis-stated definition of R in Theorem 1, which the reader also noted. Because the intended claim is correct and the defect is presentational, the conditional verdict remains appropriate.","tokens_in":5147,"tokens_out":37064,"duration_ms":348935,"concrete_test":"Symbolically Taylor-expand the centered and scaled R under both definitions for a representative case, e.g., r=1, s=2, m/n→α=1, and compare the first-order limiting variance with the theorem's formula. The intended definition X^s/(X+Y)^r reproduces the printed variance; the printed definition X^r/(X+Y)^s does not, confirming the exponent swap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing mathematical concern identified for the intended result R=X^s/(X+Y)^r. The first-order Taylor expansion is consistent with all three variance formulas, and the remainder bound (3) holds on the high-probability event A for fixed r,s>0 and p∈(0,1): each Hessian entry is O(n^{s-2}/(n+m)^r) and the deviation vector squared is O((n+m)log(n+m)), giving |Q| ≤ C n^{s-2} log(n+m)/(n+m)^{r-1}; the stated scalings then make the scaled remainder vanish in regimes (i)-(iii). The proof is terse but sound. The one clear defect is that Theorem 1 defines R=X^r/(X+Y)^s, while the abstract, the introduction, Eq. (1), and the centering and variance formulas all use R=X^s/(X+Y)^r. With the printed definition the theorem is false as stated (the centering exponent is swapped). This is a statement-level error, readily fixed, and it does not undermine the intended mathematical claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":5316,"tokens_out":6897,"duration_ms":69746,"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":[{"comment":"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":"Theorem 1"},{"comment":"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.","section":"Section 2, Eq. (3)"}],"minor_comments":[{"comment":"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":"Section 2, event A"},{"comment":"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":"Section 2, proof of Theorem 1"},{"comment":"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":"Section 3, Eq. (4)"},{"comment":"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.","section":"Section 2, remainder estimate"}],"recommendation":"minor_revision","confidential_remarks":"This is a short paper with a correct intended result. The statement-level error in Theorem 1 must be fixed, and the proof of the remainder bound should be expanded for checkability. The paper's scope fits a mathematical statistics journal. No concerns about novelty or authorship."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the intended result is correct and worth a referee, but Theorem 1 as printed is not the theorem they prove. The ratio in the statement is X^r/(X+Y)^s, while everything else—abstract, introduction, Eq. (1), centering and variance—uses X^s/(X+Y)^r. With the printed definition the centering exponent is swapped and the theorem is false; it is a statement-level typo, readily fixed, but it has to be fixed before publication.\n\nWhat is actually new: for fixed r,s>0, independent X~Bin(n,p), Y~Bin(m,p), they prove the normalized ratio converges to a zero-mean normal under three growth regimes: m/n→∞ with m log m / n^{3/2}→0, m/n→α∈(0,∞), and m/n→0. The variances are explicit and match the gradient of f(x,y)=x^s/(x+y)^r at the Binomial means. This genuinely extends the special case s=1, r=1/2 in [3]; the two-exponent form across regimes is not in the cited literature. The paper also honestly says it does not cover the motivating m∼n^2 case, which is the right call.\n\nThe proof is a Taylor expansion plus Slutsky plus a concentration bound; nothing fancy, but it works. The remainder bound (3) is asserted tersely—the constant is implicit and the Hessian estimate is one line—but the stress-test note fills in the details and it holds for fixed r,s>0 and p∈(0,1). I would still ask the authors to expand that paragraph, because as written the reader has to do real work to verify uniformity.\n\nThe simulations are illustrative only: no code or data, and the KL-divergence comparison is sensible but not a formal check. That is fine for this kind of paper.\n\nWho benefits: people working on ratios of Binomial or count variables who need a clean CLT. It is a modest, honest contribution, not a breakthrough. I would send it to a referee—conditional on fixing the exponent swap and tightening the remainder bound.","headline":"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.","tokens_in":5821,"tokens_out":2732,"would_cite":false,"duration_ms":28950,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","62E20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["ratio of binomial random variables","normal approximation","limiting distribution","Taylor expansion remainder","Slutsky lemma","asymptotic variance","simulation study"],"falsifier":"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 $x=0$.","tokens_in":4943,"feed_emoji":"📊","tokens_out":11341,"duration_ms":118785,"temperature":0.7,"pith_summary":"The paper's aim is to show that the ratio $R=X^s/(X+Y)^r$, built from two independent Binomial counts $X\\sim\\mathrm{Binomial}(n,p)$ and $Y\\sim\\mathrm{Binomial}(m,p)$, is asymptotically Normal when $n$ and $m$ both grow large, provided the growth of $m$ relative to $n$ falls into one of three regimes. In each regime the paper gives the exact centering $n^s/(n+m)^r p^{s-r}$, the exact scaling, and the exact limiting variance, all as explicit functions of $r,s,p$ and of the limiting ratio $\\alpha=\\lim m/n$. A sympathetic reader would care because ratios of counts of this shape appear in applied probability and network statistics, and a validated Normal approximation turns such ratios into usable tools for confidence intervals and tests. The paper supports the theorem with simulations across the parameter ranges.","feed_headline":"Binomial ratio X^s/(X+Y)^r is Gaussian in the limit","feed_subtitle":"For three regimes of m/n, the centered, scaled ratio becomes Normal with explicit variance — simulations confirm.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies Lemma 1, the convergence lemma that lets a term converging to zero in probability be dropped from a convergent sequence; this is what allows the scaled remainder to be ignored in the limit.","marker":"[1]"},{"why":"Supplies the eigenvalue-localization theorem and the norm inequality used to bound the Hessian and hence the quadratic remainder $Q(X,Y)$.","marker":"[2]"},{"why":"Provides the motivating case $X/\\sqrt{X+Y}$ with $m$ of order $n^2$, whose Normal limit the present paper generalizes to $X^s/(X+Y)^r$.","marker":"[3]"}],"fun_headline_variants":["Gaussian limit proved for binomial ratios","Three regimes, one normal limit for binomial ratio","Centered binomial ratio converges to normal distribution","Explicit variance found for binomial ratio limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Gaussian limit proved for binomial ratios","Three regimes, one normal limit for binomial ratio","Centered binomial ratio converges to normal distribution","Explicit variance found for binomial ratio limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000379,"raw_usage":{"total_tokens":1968,"prompt_tokens":855,"completion_tokens":1113,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":471,"completion_tokens_details":{"reasoning_tokens":1058}},"tokens_in":471,"tokens_out":1113,"duration_ms":11958,"temperature":1.0,"reasoning_tokens":1058,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:37:52.850607+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $x=0$.","supporting_citations":[{"cited_title":"Random line graphs and edge-attributed network inference","cited_arxiv_id":"2103.14726","evidence_quote":"Provides the motivating case $X/\\sqrt{X+Y}$ with $m$ of order $n^2$, whose Normal limit the present paper generalizes to $X^s/(X+Y)^r$."}],"review_version":1}