{"id":"cf897b0c-b4e3-4379-b2bc-2a814bd621e6","arxiv_id":"2411.15861","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A CLT is proved for general linear spectral statistics of high-dimensional Spearman rank correlation matrices, including a new CLT for the improved Spearman matrix.","lead":"This paper proves a central limit theorem for linear spectral statistics of Spearman rank correlation matrices when the number of variables grows with the sample size, extending earlier work that covered only polynomial statistics. It also gives the first such limit theorem for Hoeffding's improved Spearman matrix, a U-statistic of order 3, and uses both to build independence tests.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main CLT depends on two key estimates from unpublished Bao (2019b): Lemma A.2 (quadratic-form concentration) and Lemma 6.1 (edge rigidity); without a published proof, Theorem 3.2 remains conditional.","rationale":"The reader's weakest-assumption analysis correctly identified Lemma 6.1, the edge-rigidity bound imported from Bao (2019b), as a load-bearing external input. The stress-test pass confirms this and finds that the same unpublished preprint is also the source of Lemma A.2, the concentration inequality for quadratic forms that underpins the covariance computation in Lemma 6.2. Both estimates are used essentially: Lemma A.2 controls the martingale-difference terms and the covariance of quadratic forms in Step 1, while Lemma 6.1 controls the truncation error (20) and provides the uniform spectral-norm moment bounds (32) needed for tightness and for the contour-integral identity (18). Without published versions or self-contained proofs of these two ingredients, the main theorem is conditional on the validity of an external, not-yet-available result. This does not demonstrate an internal contradiction, and the paper's simulation results and its consistency with Bao et al. (2015) for polynomial statistics provide supporting, though not dispositive, evidence. The reader's CONDITIONAL verdict is therefore appropriate and no adjustment is needed; the concern is best addressed by asking the authors to supply the missing proofs or a published reference.","tokens_in":51,"tokens_out":35241,"duration_ms":399427,"concrete_test":"Ask the authors to append a self-contained proof of Lemma A.2 and Lemma 6.1, or to cite a published version of Bao (2019b). As a focused analytical check, re-derive Lemma A.2 from the exact covariance structure in Lemma A.1 and the exchangeability of the rank vector, without invoking Proposition 2.1 of Bao (2019b); if the n^{-q/2+δ} concentration cannot be established, then the covariance computation in Step 1 of Lemma 6.2 is unsupported and the CLT is conditional on an unverified external input.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 3.2) is obtained by integrating the Gaussian-process limit of the Stieltjes transform in Lemma 6.2. Lemma 6.2 in turn rests on two quantitative probabilistic inputs imported from the unpublished preprint Bao (2019b). First, Lemma A.2 uses Proposition 2.1 of Bao (2019b) to prove the concentration bound (54) for arbitrary moments of centered quadratic forms in the standardized rank vector s. This bound is essential in Step 1 of Lemma 6.2 to control the martingale differences Y_j(z) and to evaluate the covariance limit (24); without it, the explicit covariance formula involving the terms 2tr(AB) - (6/5)tr(A∘B) - (4/(5n))tr(A)tr(B) is not justified. Second, Lemma 6.1, the edge-rigidity estimate (19), is asserted in Remark 6.1 to follow directly from Proposition 2.3 of Bao (2019b). This estimate is used to justify the contour integration in (18), to control the truncation error (20), and to obtain the uniform moment bounds in (32) on which the tightness argument for M_n(z) depends. If either of these estimates fails, or if Bao (2019b) imposes extra conditions beyond the stated \"doubly independent and absolutely continuous\" assumptions, then the finite-dimensional convergence and tightness of M_n(·) do not close, and Theorems 3.1-3.3 do not follow. This is a verifiability and robustness concern: the paper makes the dependence explicit but supplies no proof and no published surrogate.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper establishes a central limit theorem for linear spectral statistics (LSS) of Spearman's rank correlation matrices in the proportional-growth regime p/n -> y. The proof represents the rank vectors as columns with a permutation-uniform distribution, forms the associated Gram matrix, and applies a Bai--Silverstein martingale decomposition to the Stieltjes transform. The resulting Gaussian-process limit for analytic test functions generalizes the polynomial LSS result of Bao et al. (2015). The same framework is applied to Hoeffding's improved Spearman matrix, a U-statistic of order 3; the paper shows that replacing the classical Spearman matrix by the improved version changes only the asymptotic mean. These results are used to construct four tests of independence, with simulations for normal, Cauchy, and mixed distributions.","tokens_in":33530,"tokens_out":7269,"duration_ms":65500,"significance":"If the theorems are correct, the paper fills a natural gap: it upgrades polynomial LSS CLTs for Spearman matrices to analytic LSS, and it provides what appears to be the first CLT for LSS of a matrix that is a U-statistic of order 3. The covariance computation is nontrivial because the rank vectors are not independent and have a singular covariance; the three-term covariance formula in Lemma A.1 is explicit, and the polynomial limit is checked against the known result of Bao et al. (2015). The proposed tests are clearly motivated and the simulations support their usefulness under heavy tails. The main reservations are verifiability: two quantitative estimates that carry the proof are imported from an unpublished preprint, and one further approximation lemma is delegated to prior papers by reference only.","major_comments":[{"comment":"The edge-rigidity bound (19) is load-bearing: it is used to justify the contour-integral representation (18), the truncation error estimate (20), and the uniform moment bound (32) on which the tightness of M_n(z) depends. However, Lemma 6.1 is not proved in the manuscript; Remark 6.1 asserts it follows directly from Proposition 2.3 of Bao (2019b), which is listed as an unpublished preprint. Consequently Theorems 3.1--3.3 are conditional on a result that is neither proved here nor available in a published refereed form. I ask the authors to provide a complete proof of Lemma 6.1 in the appendix, or to replace the reference by a published version, and to state explicitly whether Proposition 2.3 of Bao (2019b) requires assumptions beyond 'doubly independent and absolutely continuous entries' (for example, moment or smoothness conditions). The theorem statements should be amended if the imported result has extra hypotheses.","section":"Section 6.1, Lemma 6.1 and Remark 6.1"},{"comment":"The concentration inequality (55), used at multiple points in Steps 1 and 3 of Lemma 6.2 (e.g., to prove (23) and to justify the limits (27)--(30)), is imported from Proposition 2.1 of the same unpublished Bao (2019b). Since the bound is needed in the exact form stated, with exponent n^{-q/2+delta} and operator norms, a proof or a precise published citation is required. Without this bound, the martingale-difference verification of finite-dimensional convergence and the explicit covariance formula in Theorem 3.1 do not close. This is a separate load-bearing input from Lemma 6.1.","section":"Appendix, Lemma A.2"},{"comment":"The proof of Lemma 6.4 consists of a one-sentence reference to Wu and Wang (2022) and Li et al. (2023), without theorem or equation numbers, and those papers do not appear to state the Frobenius-norm bounds (50)--(51) in this form. Lemma 6.4 is essential for replacing ~rho_n and K_n by U_n and V_n in the derivation of Lemma 6.3; the bound E ||~rho_n - U_n||_F^2 = o(p) is nontrivial and should not be delegated without precise pointers. Please include a complete proof or state exactly where in the cited papers these estimates are proved.","section":"Section 6.3, Lemma 6.4"}],"minor_comments":[{"comment":"The sentence 'they utilized ... and proposed a two-step comparison approach' contains a duplicated 'for for any positive integer k'; please correct the typo.","section":"Section 3.2, Remark 3.2"},{"comment":"The notation for the Stieltjes transform of rho_n/y_n is not visually distinguished from that of g_n in the displayed equations; please introduce separate symbols or a clear subscript so the two transforms can be told apart.","section":"Section 3.1, after equation (6)"},{"comment":"The condition 'as y_n -> y' should be stated as 'as p/n -> y' for consistency with Theorem 3.2, since y_n = p/n in the paper's notation.","section":"Section 3.3, Theorem 3.3"},{"comment":"The notation lambda_min{n,p}(rho_n) should be defined explicitly: if interpreted as the smallest eigenvalue of rho_n, the statement is false for p > n because rho_n then has p-n zero eigenvalues; presumably the index min(n,p) in the descending order (i.e., the smallest nonzero eigenvalue when p > n) is intended. Please clarify.","section":"Section 6.1, Lemma 6.1"},{"comment":"There are several typographical errors, including 'matirx', 'neighborhoog', 'rouine', and 'concerntration'; these should be corrected in a revision.","section":"Section 4 and Section 5"}],"recommendation":"major_revision","confidential_remarks":"The central theorems are potentially important, but the dependence on the unpublished preprint Bao (2019b) for both Lemma 6.1 and Lemma A.2 is the main obstacle to publication in the current form. I did not find evidence of circularity: the polynomial reduction to Bao et al. (2015) is used as a consistency check rather than as an input, and the improved Spearman mean shift is derived from a matrix difference. If the authors can supply complete proofs of the two imported estimates (or replace them with published sources), and a full proof of Lemma 6.4, I would be willing to reconsider. The numerical section is not central to the theoretical claim but supports the applications."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core of this paper is a genuine advance: it proves a CLT for general analytic linear spectral statistics of Spearman's rank correlation matrices, going beyond the polynomial functions of Bao et al. (2015), and it gives the first CLT for a standard U-statistic of order 3 by handling Hoeffding's improved Spearman matrix. The Bai–Silverstein martingale machinery is a good fit, and the covariance computation is the real substance—the three leading terms in the quadratic-form covariance are derived explicitly, and the polynomial case reduces to Bao et al. as a consistency check. That check is clean.\n\nThe main soft spot is dependence on two estimates from the unpublished preprint Bao (2019b): Lemma 6.1 (edge rigidity for Spearman eigenvalues) and Lemma A.2 (concentration of centered quadratic forms in the standardized rank vector). The paper is upfront about this—Remark 6.1 cites Proposition 2.3 of Bao (2019b) and Lemma A.2 cites Proposition 2.1—but a referee cannot verify either from the manuscript. These are not decorative: the rigidity bound drives the contour integration and tightness, and the concentration bound is what makes the martingale differences vanish. If either hides extra conditions beyond the stated 'doubly independent, absolutely continuous' assumptions, Theorem 3.2 needs adjustment.\n\nThe simulation study is useful but thin on reporting: no standard errors or confidence intervals are given, so the empirical sizes and powers are point estimates from 1000 replications. That is a minor fix. There are also a handful of typos ('neighborhoog', 'matirx') that should be caught in copy-editing.\n\nI found no circularity or fitted parameters. The polynomial consistency check is an independent verification, not an input. The citation practice is broadly appropriate, and citing a preprint is a verifiability issue rather than a sign of carelessness.\n\nThis paper is for people working on high-dimensional nonparametric correlation and independence testing, and for RMT folks interested in U-statistics of order higher than 2. It deserves serious peer review. Send it out, but ask the authors to supply proofs or published surrogates for Lemma 6.1 and Lemma A.2, and to add a measure of simulation variability.","headline":"A technically strong CLT paper that generalizes Bao et al. (2015) to analytic LSS and adds the first order-3 U-statistic CLT, but the main theorem depends on two unproven lemmas imported from an unpublished Bao preprint.","tokens_in":34064,"tokens_out":3883,"would_cite":true,"duration_ms":33047,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","62H15","62G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a central limit theorem for linear spectral statistics of large Spearman rank correlation matrices, covering analytic functions and yielding the first CLT for Hoeffding's improved Spearman matrix.","keywords":["Spearman rank correlation matrix","linear spectral statistics","random matrix theory","central limit theorem","U-statistics","improved Spearman","Marchenko-Pastur law","independence testing"],"falsifier":"Simulate $n\\times p$ doubly independent absolutely continuous data with $p/n$ fixed and examine whether $P(\\lambda_1(\\rho_n)>\\eta_r)$ and $P(\\lambda_{\\min\\{n,p\\}}(\\rho_n)\\le\\eta_l)$ decay faster than any power of $n$ for $\\eta$ outside the Marchenko-Pastur support; alternatively, verify Proposition 2.3 of Bao (2019b) on which Lemma 6.1 is based. A data-generating mechanism for which the edge tails are only polynomially small would show that Lemma 6.1, and hence the proof of Theorems 3.1–3.3, does not hold as stated.","tokens_in":33000,"feed_emoji":"📊","tokens_out":5883,"duration_ms":49780,"temperature":0.7,"pith_summary":"This paper establishes a central limit theorem for the linear spectral statistics of large-dimensional Spearman rank correlation matrices when dimension and sample size grow together. It shows that for analytic functions $f$, the statistic $T(f)=p(\\int f\\,dF^{\\rho_n}-\\int f\\,dF_y)$ converges, jointly over several $f$'s, to a Gaussian vector whose mean and covariance are given by explicit contour integrals. This extends an earlier polynomial-only CLT to functions such as the logarithm of the determinant. The same method yields a CLT for Hoeffding's improved Spearman matrix, a standard U-statistic of order 3, which is the first such result. These CLTs feed new tests for mutual independence whose empirical sizes stay near nominal level under heavy-tailed data.","feed_headline":"Spearman rank matrices get a Gaussian limit at every analytic statistic","feed_subtitle":"New theorem covers determinants and powers, extends polynomial results, and powers heavy-tailed independence tests.","key_machinery":"The central object is the Gram matrix $g_n = \\frac{1}{p}\\sum_j s_j s_j^{\\top}$ of the standardized rank vectors, whose nonzero eigenvalues coincide with those of the Spearman matrix scaled by $p/n$. Because ranking makes the columns dependent, the paper treats them as a sample from a population with covariance $\\Sigma = \\frac{n}{n-1}(I_n - \\frac{1}{n}\\mathbf{1}\\mathbf{1}^{\\top})$ and uses the Stieltjes transform together with a martingale decomposition. The load-bearing identity is the covariance formula for quadratic forms in one column, $\\operatorname{cov}(s^{\\top}As, s^{\\top}Bs)=2\\operatorname{tr}(AB) - \\frac{6}{5}\\operatorname{tr}(A\\circ B) - \\frac{4}{5n}\\operatorname{tr}(A)\\operatorname{tr}(B)+O(\\|A\\|\\|B\\|)$, whose three leading terms produce the three corrections in the Gaussian mean and covariance. A strong edge-rigidity estimate, imported from an unpublished source, is used to keep the extreme eigenvalues inside the integration contour.","core_discovery":"The paper's central claim is Theorem 3.2: under doubly independent, absolutely continuous entries and $p/n\\to y\\in(0,\\infty)$, the process $\\{T(f)\\}$ over analytic $f$ converges weakly to a Gaussian process with mean $EZ_f = -\\frac{1}{2\\pi i}\\oint f(z)\\frac{\\mu(z/y)}{y}\\,dz$ and covariance $\\operatorname{cov}(Z_f,Z_g) = -\\frac{1}{4\\pi^2}\\oint\\!\\oint f(z_1)g(z_2)\\frac{\\sigma(z_1/y,z_2/y)}{y^2}\\,dz_1 dz_2$. The mean and covariance are built from the Stieltjes transform $s(z)$ of the limiting Marchenko-Pastur law and from three covariance corrections specific to rank vectors: the standard $2\\operatorname{tr}(AB)$ term, a Hadamard-product term $-\\frac{6}{5}\\operatorname{tr}(A\\circ B)$, and a new trace-product term $-\\frac{4}{5n}\\operatorname{tr}(A)\\operatorname{tr}(B)$. Theorem 3.3 shows the same covariance, with an extra mean term, for Hoeffding's improved Spearman matrix, obtained by controlling the difference between the classical and improved rank matrices.","pith_inferences":["The covariance identity with coefficients $6/5$ and $4/5$ suggests an exchangeable-rank universality class: any rank-based Gram matrix built from vectors uniform on permutations may have LSS CLTs with the same Gaussian covariance, differing only in mean shifts; Kendall's matrix, being a U-statistic of order 2, is a natural test case for this pattern.","Remark 3.4 suggests replacing the ratio $p/n$ by $p/(n-1)$ in the centering; a direct simulation comparing empirical sizes of the proposed tests under both centering choices would show whether the extra mean term $\\mu_3$ is practically removable, a testable extension.","If the imported edge-rigidity bound were replaced by a proven edge bound under weaker moment assumptions, the CLT could extend to heavier-tailed or discrete data; numerical checks of the tail probability $P(\\lambda_1(\\rho_n)>\\eta_r)$ could indicate how strong an assumption is really needed."],"forward_implications":["Theorem 4.1 gives explicit CLTs for $\\log|\\rho_n|$ and $\\operatorname{tr}(\\rho_n^k)$ and for their improved-Spearman analogues; for example, $\\log|\\rho_n| + (n-p)\\log(1-y_n)+p$ is asymptotically normal with mean $\\frac{3}{2}\\log(1-y)+2y$ and variance $-2\\log(1-y)-2y$.","The improved Spearman CLT is the first CLT for linear spectral statistics of a standard U-statistic of order 3 in random matrix theory.","The resulting tests $L_{\\rho,2}$, $L_{\\rho,\\log}$, $L_{\\tilde{\\rho},2}$, and $L_{\\tilde{\\rho},\\log}$ have empirical sizes close to the nominal 5% level under normal, Cauchy, and mixed distributions, whereas Pearson-correlation tests fail completely under heavy tails.","The polynomial results reproduce the earlier results of Bao et al. (2015), confirming that the Stieltjes-transform route and the moment and cumulant route agree where both apply."],"supporting_citations":[{"why":"Supplies the Stieltjes-transform CLT method, the martingale decomposition, and the contour-integration framework used throughout the proof.","marker":"Bai and Silverstein (2004)"},{"why":"Establishes the polynomial-function CLT for Spearman matrices that this paper extends to analytic functions.","marker":"Bao et al. (2015)"},{"why":"Introduces the improved Spearman correlation as a standard U-statistic of order 3, the object of Theorem 3.3.","marker":"Hoeffding (1948)"},{"why":"Provides the limiting spectral distribution of Spearman matrices and the Hoeffding-type decomposition used in Lemma 6.3.","marker":"Wu and Wang (2022)"},{"why":"Gives the Kendall's tau CLT and the edge rigidity of Kendall matrices used to control the difference between classical and improved Spearman matrices.","marker":"Li et al. (2021)"},{"why":"Source of the Hadamard-product correction $\\operatorname{tr}(A\\circ B)$ in the covariance of quadratic forms, which enters the Gaussian mean and covariance.","marker":"Pan and Zhou (2008)"},{"why":"The unpublished preprint whose Proposition 2.3 is cited for the strong edge-rigidity bound in Lemma 6.1 and for the concentration inequality in Lemma A.2.","marker":"Bao (2019b)"},{"why":"Establishes the Marchenko-Pastur limit for the Gram matrix of Spearman rank statistics, the starting point for the CLT.","marker":"Bai and Zhou (2008)"}],"fun_headline_variants":["Spearman rank matrices get CLT for all analytic statistics","New CLT for Spearman rank correlation spectral statistics","Gaussian law for Spearman rank matrices' linear statistics","Spearman CLT: new covariance terms enable independence tests","Rank correlation spectral CLT with explicit covariance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof requires a very strong concentration bound for the largest and smallest eigenvalues of the Spearman matrix (tails of order $o(n^{-m})$ for every $m$), which is cited to an unpublished preprint and not proved in the paper; if that bound fails, the Gaussian-process convergence is not established.","fun_headline_variants_meta":{"raw":{"variants":["Spearman rank matrices get CLT for all analytic statistics","New CLT for Spearman rank correlation spectral statistics","Gaussian law for Spearman rank matrices' linear statistics","Spearman CLT: new covariance terms enable independence tests","Rank correlation spectral CLT with explicit covariance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000937,"raw_usage":{"total_tokens":3999,"prompt_tokens":928,"completion_tokens":3071,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":544,"completion_tokens_details":{"reasoning_tokens":2994}},"tokens_in":544,"tokens_out":3071,"duration_ms":20943,"temperature":1.0,"reasoning_tokens":2994,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:49:08.068903+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate $n\\times p$ doubly independent absolutely continuous data with $p/n$ fixed and examine whether $P(\\lambda_1(\\rho_n)>\\eta_r)$ and $P(\\lambda_{\\min\\{n,p\\}}(\\rho_n)\\le\\eta_l)$ decay faster than any power of $n$ for $\\eta$ outside the Marchenko-Pastur support; alternatively, verify Proposition 2.3 of Bao (2019b) on which Lemma 6.1 is based. A data-generating mechanism for which the edge tails are only polynomially small would show that Lemma 6.1, and hence the proof of Theorems 3.1–3.3, does not hold as stated.","supporting_citations":[],"review_version":1}