{"id":"0ceadb43-4f33-4f06-b3a3-9cc03d3f4409","arxiv_id":"2501.08979","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A claimed Berry-Esseen bound of log^{5/4}(d)/n^{1/8} for high-dimensional self-normalized sums, with proof gaps in the covariance comparison and corollary derivations.","lead":"This paper gives a new quantitative bound for how close the largest of many self-normalized sums is to a Gaussian distribution in high dimensions. The advertised rate is slow (a fractional power of sample size) and the proof has several invalid steps in key lemmas, so the main theorem is not established as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 1.1's proof uses a false normalization (variance 1 instead of 1/n), so the claimed lower bound \"≥1/2\" is impossible and the advertised n^{-δ/8} rate for Δ_n is not established.","rationale":"Reader's REJECT is appropriate. I read the paper as aiming to establish explicit Berry-Esseen rates for self-normalized sums; the abstract and Corollary 1.1 center on Δ_n. The reader's first cited flaw (Lemma 5) is real but only affects the Δ^X_n part of Theorem 1. The false lower bound in Corollary 1.1 is more load-bearing because it is used exactly where the advertised Δ_n rate must handle small truncation levels a_j. The identity in the display confuses total variance 1 with 1/n; with Lemma 2's equality E[(X_j)^2 1{(X_j)^2≤a_j^2 n}]=a_j^2, the LHS of the display is 1 - a_j^2/(σ_j^2 n), not the tail expectation. The asserted tail expectation ≥1/2 cannot hold since the full expectation is 1/n. This is a concrete internal inconsistency, not a style or literature objection. The theorem's Δ_n bound might be recoverable by a different argument, but the submitted proof does not establish it. Since this is the same central defect the reader flagged (though I weight the Corollary issue higher than the Lemma 5 issue), the verdict remains REJECT; no read adjustment.","tokens_in":24591,"tokens_out":12122,"duration_ms":116490,"concrete_test":"Verify the normalization in the Corollary 1.1 display: with X1~N(0,1), σ=1, n=4, compute E[(X1/2)^2 1{X1^2>2}] ≈ 0.144, contradicting the claimed lower bound ≥1/2. Equivalently, derive from E[(X_j/(σ_j n^{1/2}))^2]=1/n the universal bound E[(X_j/(σ_j n^{1/2}))^2 1{X_j^2>σ_j^2 n/2}] ≤ 1/n, which is <1/2 for all n>2. If the identity is still in question, redo the case a_j^2=σ_j^2/4 with Lemma 2 and track the factor n^{-1} in the truncated second moment.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The advertised rate for Δ_n is Corollary 1.1. Its proof hinges on the case a_j^2 ≤ σ_j^2/2 and on the display 1 - E[(e_j^T X1/(σ_j n^{1/2}))^2 1{(e_j^T X1)^2 ≤ a_j^2 n}] = E[(e_j^T X1/(σ_j n^{1/2}))^2 1{(e_j^T X1)^2 > a_j^2 n}]. This identity is false: E[(e_j^T X1/(σ_j n^{1/2}))^2] = 1/n, not 1. Using Lemma 2, the left-hand side is 1 - a_j^2/(σ_j^2 n), which is near 1 for large n, whereas the right-hand side is a tail expectation of order at most 1/n. The subsequent claim that E[(e_j^T X1/(σ_j n^{1/2}))^2 1{(e_j^T X1)^2 > σ_j^2 n/2}] ≥ 1/2 is therefore impossible: the quantity is bounded by 1/n for every distribution. This \"≥1/2\" is the only device used to handle a_j^2 ≤ σ_j^2/2 via the trivial bound Δ_n ≤ 1. Removing it leaves the terms of Theorem 1 with denominators a_j unaccounted for by moments of X_j/σ_j. This breaks the proof of the n^{-δ/8} log^{5/4}(ed) rate, the paper's headline result. The Gaussian comparison issue in Lemma 5 is also real, but it concerns Δ^X_n and is secondary to this defect in the main Δ_n bound.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the coordinate-wise maximum of self-normalized sums T_n defined by e_j^T T_n = |\\sum_i e_j^T X_i| / (\\sum_i (e_j^T X_i)^2)^{1/2}, with X_1,...,X_n iid centered in R^d. It proposes two Gaussian approximation distances: Delta_n, the best approximation by a centered Gaussian with correlation-matrix covariance, and Delta_n^X, the approximation by the Gaussian with covariance Corr(X_1). Theorem 1 gives explicit bounds for both in terms of truncation levels a_j, a tail probability, and truncated moment terms. Corollary 1.1 claims that under E[max_j |e_j^T X_1/sigma_j|^{2+delta}] < infinity for delta in (0,1], Delta_n is bounded by C log^{5/4}(ed) n^{-delta/8} (E max_j |e_j^T X_1/sigma_j|^{2+delta})^{1/4}, giving n^{-1/8} when third moments are finite. The proof follows the Bentkus-Gotze strategy via smooth approximation of the indicator of a maximum, Gaussian comparison, and a refined high-dimensional CLT stated as Proposition 1.","tokens_in":24952,"tokens_out":14038,"duration_ms":128943,"significance":"If valid, the paper would be a genuine advance: it would give the first explicit high-dimensional Berry-Esseen bound for self-normalized coordinatewise maxima under only finite (2+delta)-th moments of the standardized maximum, with d growing faster than n. The paper is clearly organized, and Proposition 1 is a useful-looking refinement of the high-dimensional CLT of Chernozhukov et al. (2022), separating the fourth-moment scale B_n from the sup-norm scale D_n. There is no indication of circularity or parameter fitting: the arguments are derivations from published inequalities. However, two load-bearing points are incorrect: the proof of Corollary 1.1 uses a false normalization identity, and Lemma 5 replaces an independent Gaussian vector by a function of X_1. These errors invalidate the advertised rate and the stated bound for Delta_n^X, so the central claims of the manuscript are not presently supported.","major_comments":[{"comment":"The displayed identity at the start of the proof is false. By Lemma 2, E[(e_j^T X_1)^2 1{(e_j^T X_1)^2 <= a_j^2 n}] = a_j^2, so 1 - E[(e_j^T X_1/(sigma_j n^{1/2}))^2 1{(e_j^T X_1)^2 <= a_j^2 n}] equals 1 - a_j^2/(sigma_j^2 n), not the tail expectation written on the right-hand side; equivalently, E[(e_j^T X_1/(sigma_j n^{1/2}))^2] = 1/n, not 1. Consequently the subsequent claim that E[(e_j^T X_1/(sigma_j n^{1/2}))^2 1{(e_j^T X_1)^2 > sigma_j^2 n/2}] >= 1/2 is impossible, since this expectation is at most 1/n for every distribution. This lower bound is the only device used to handle the case a_j^2 <= sigma_j^2/2; without it, the terms in Theorem 1 that involve denominators a_j are not controlled by the assumed (2+delta)-th moments of e_j^T X_1/sigma_j. The advertised rate Delta_n <= C log^{5/4}(ed) n^{-delta/8} (...) is therefore not established.","section":"4, Proof of Corollary 1.1"},{"comment":"Equation (15) splits |E[Z^X_{1,j} Z^X_{1,k}] - E[Y_{1,j} Y_{1,k}]| by inserting the event E_{j,k}, but the proof then treats the independent Gaussian pair (Z^X_{1,j}, Z^X_{1,k}) as if it were equal to (X_{1,j}/sigma_j, X_{1,k}/sigma_k) on E_{j,k}. This is invalid because Z^X_1 is independent of X_1. The first term should be E[Z^X_{1,j} Z^X_{1,k}] P(E_{j,k}) - E[X_{1,j} X_{1,k} 1(E_{j,k})]/(n a_j a_k), not the displayed difference of cross-moments. For j=k, this quantity equals P((e_j^T X_1)^2 <= a_j^2 n) - 1/n, which can be close to 1, whereas the bound in Eq. (16) can be near 0 when a_j is close to sigma_j. Thus the bound varpi_n <= 6 R_n is unsupported, and the stated bound for Delta_n^X is not proven. This does not by itself invalidate the Delta_n half of Theorem 1, whose proof compares only with Z having covariance Var(Y_i), but it invalidates the moment-matching Gaussian approximation result claimed in Theorem 1.","section":"4.1, Lemma 5, Eq. (15)"}],"minor_comments":[{"comment":"The proof uses p for the dimension in the union bound, although the paper uses d throughout; p should be replaced by d.","section":"4.1, Lemma 3"},{"comment":"In the case sigma_j^2/2 > a_j^2, the displayed denominator a_j in E[(e_j^T X_1)^2/a_j 1{...}] should presumably be a_j^2, for dimensional consistency with the surrounding tail terms.","section":"4.1, Lemma 5"},{"comment":"The condition 'nE[||Y_1||_infty]^3 <= 1' appears to be a typo for 'nE[||Y_1||_infty^3] <= 1'; the proof and the stated bound both use E[||Y_1||_infty^3].","section":"4.1, Lemma 6"},{"comment":"Equation (31) states Delta_1 <= n^{-1} B_n^2, but from (28) one immediately gets Delta_1 <= b_2^4 B_n^4/n; the displayed bound seems to be a typo, although the subsequent algebra uses the B_n^4 form.","section":"4.2, Proposition 1, Eq. (31)"}],"recommendation":"reject","confidential_remarks":"The manuscript's central results are not currently correct: the proof of Corollary 1.1 relies on a false identity and an impossible lower bound, and Lemma 5, which supports the Delta_n^X bound in Theorem 1, makes an invalid independence substitution. I see no indication of circularity or parameter-fitting, and the refinement in Proposition 1 may be of independent interest, but the advertised main rate is not established. For a journal whose standard is correctness of the central claims, these are load-bearing errors that cannot be repaired by local rewriting within the current proof strategy."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper addresses a real gap: a Berry-Esseen bound for the coordinate-wise maximum of self-normalized sums without Das's independence-across-coordinates assumption. The proof idea—combining Bentkus–Götze's self-normalized technique with modern high-dimensional CLT tools—is sensible, and the technical lemmas (6–8) are substantial. I believe the main theorem, as an abstract bound in terms of the truncated moments a_j, is likely correct.\n\nBut the advertised rate, the n^{-δ/8} log^{5/4}(ed) bound in Corollary 1.1, is not established. The proof contains a normalization error: it claims 1 - E[(e_j^T X1/(σ_j n^{1/2}))^2 1{(e_j^T X1)^2 ≤ a_j^2 n}] equals the tail expectation E[(e_j^T X1/(σ_j n^{1/2}))^2 1{(e_j^T X1)^2 > a_j^2 n}], but the left side is 1/n minus that tail, not 1 minus the tail. As a result, the displayed lower bound \"≥1/2\" on a quantity that is at most 1/n is impossible. That lower bound is the only mechanism for treating the case a_j^2 ≤ σ_j^2/2, so the corollary's rate collapses.\n\nThere is a second problem, in Lemma 5, for the moment-matching Gaussian approximation Δ^X_n. The proof of equation (15) treats the independent Gaussian vector Z^X as if, on the truncation event, it were equal to X1/(n^{1/2}σ) for the purpose of the cross-moment. But Z^X is independent of X1; its conditional cross-moment is the constant correlation, not the random product. So the covariance comparison bound is not proven as written. This is less central for the Δ_n bound, but it matters for the paper's claims about Δ^X_n.\n\nThese are not cosmetic gaps. The headline rate is broken and one lemma's proof is invalid. On the other hand, the paper is not a mess: the proof strategy is sound in outline, the literature is handled honestly, and the flaws may be repairable. The main theorem might survive a corrected Lemma 5 (or a different argument), and the corollary might be fixable by a more careful truncation argument.\n\nWho should read this? Researchers working on high-dimensional self-normalized CLTs and bootstrap procedures. It deserves a serious referee—the errors are subtle enough that a good referee is needed—but the current version should not be accepted as is. I'd advise major revision with a request to prove or correct Corollary 1.1 and Lemma 5, not a desk rejection.","headline":"A serious paper with a genuinely new theorem, but the advertised rate is broken by a false normalization in Corollary 1.1 and an invalid coupling in Lemma 5.","tokens_in":25480,"tokens_out":5001,"would_cite":false,"duration_ms":46966,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","60G50","62E20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims an explicit Berry–Esseen rate of $\\log^{5/4}(ed)n^{-1/8}$ for maximal self-normalized sums, obtained by truncation and Gaussian comparison.","keywords":["Berry-Esseen bound","self-normalized sums","high-dimensional central limit theorem","Gaussian approximation","coordinate-wise maximum","heavy tails","Kolmogorov-Smirnov distance","truncated moments"],"falsifier":"To test the key step, choose a centered distribution with a single coordinate pair whose truncation event is not independent of the product $e_j^\\top X_1\\,e_k^\\top X_1$, compute the true value $E[e_j^\\top Z_1^X e_k^\\top Z_1^X 1\\{E_{j,k}\\}] = E[e_j^\\top Z_1^X e_k^\\top Z_1^X]P(E_{j,k})$ and compare it with the paper's replacement $E[(e_j^\\top X_1e_k^\\top X_1/(n\\sigma_j\\sigma_k))1\\{E_{j,k}\\}]$; a numerical difference here means Lemma 5's covariance bound, and with it the stated rate, does not follow from the given proof.","tokens_in":24374,"feed_emoji":"📊","tokens_out":17832,"duration_ms":156845,"temperature":0.7,"pith_summary":"This paper tries to establish a Berry–Esseen bound for the coordinate-wise maximum of self-normalized sums in high dimensions: how close is the distribution of $\\|T_n\\|_\\infty$ to the distribution of the maximum of a Gaussian vector, measured in Kolmogorov–Smirnov distance. The main theorem bounds this distance by a tail term plus polynomial combinations of truncated first and third moments of the observations, and the corollary converts that into an explicit rate: under a finite $(2+\\delta)$-th moment of the standardized maximum, the error is $C \\log^{5/4}(ed)\\, n^{-\\delta/8}$, so with finite third moments the error goes to zero whenever $\\log d = o(n^{1/10})$. A sympathetic reader would care because these are exactly the statistics used for Studentized inference, and no previous high-dimensional CLT for their maximum was available without strong structural assumptions such as independent coordinates. The paper also argues that the best Gaussian approximation can converge even when the second moment of $X_1$ diverges, which the moment-matching Gaussian approximation cannot capture.","feed_headline":"Self-normalized sums get an explicit n^{-1/8} Gaussian bound","feed_subtitle":"Under finite third moments the maximum of Studentized coordinate sums is within C log^{5/4}(ed) n^{-1/8} of Gaussian.","key_machinery":"The load-bearing object is the truncated vector $Y_i$ defined coordinate-wise by $e_j^\\top Y_i = (e_j^\\top X_i/(a_j n^{1/2}))\\,1\\{(e_j^\\top X_i)^2\\le a_j^2 n\\}$, where $a_j$ is the largest value solving $E[(e_j^\\top X_1)^2 1\\{(e_j^\\top X_1)^2\\le a_j^2 n\\}]=a_j^2$; this makes $E[(e_j^\\top Y_i)^2]=1/n$ and $\\|Y_i\\|_\\infty\\le1$, matching the scale of a single Gaussian summand. Around these vectors the proof builds a smoothed indicator $H_{\\varepsilon,t}(x)=E[1\\{\\|x+\\varepsilon W\\|_\\infty\\le t\\}]$ with a standard Gaussian $W$, trades the self-normalized sum $T_n^Y$ for a linearized sum $\\tilde Y$ through a bounded function $g$ that replaces the reciprocal square root, and controls $|E[H_{\\varepsilon,t}(\\tilde Y)]-E[H_{\\varepsilon,t}(Y)]|$ by Taylor expansions whose derivative terms are bounded by $h_j\\lesssim \\varepsilon^{-j}(\\log d)^{j/2}$. The remaining comparison between the sum $Y$ and a Gaussian $Z$ is handled by the paper's Proposition 1, a refinement of the high-dimensional central limit theorem that separates the fourth-moment size from the $\\ell^\\infty$ size. This mechanism converts a nonlinear self-normalized object into a sum-of-independent-vectors problem, paying a $\\log^{5/4}(ed)n^{-1/8}$ price for the normalization and truncation.","core_discovery":"The central claim is Theorem 1: with $T_n$ the coordinate-wise self-normalized sum whose $j$-th entry is $|\\sum_{i=1}^n e_j^\\top X_i|/\\sqrt{\\sum_{i=1}^n (e_j^\\top X_i)^2}$, there is an absolute constant $C$ such that the best Gaussian approximation distance $\\Delta_n=\\inf_G \\sup_t |P(\\|T_n\\|_\\infty\\le t)-P(\\|Z\\|_\\infty\\le t)|$ is at most $C\\bigl(nP(\\max_j (e_j^\\top X_1)^2/a_j^2>n)+(n\\log^{5/2}(ed)\\|E[Y_1]\\|_\\infty)^{1/2}+(n\\log^5(ed)E\\|Y_1\\|_\\infty^3)^{1/4}\\bigr)$. The same theorem bounds the moment-matching distance $\\Delta_n^X$ by the same expression plus an extra truncated second-moment tail term. Under $E[\\max_j |e_j^\\top X_1/\\sigma_j|^{2+\\delta}]<\\infty$ for $\\delta\\in(0,1]$, Corollary 1.1 reduces the bound to $C\\log^{5/4}(ed)n^{-\\delta/8}(E\\max_j|e_j^\\top X_1/\\sigma_j|^{2+\\delta})^{1/4}$, so the finite-third-moment case gives $n^{-1/8}$. The proof truncates each coordinate at a level $a_j$ chosen so the truncated variables have a fixed scale, replaces the self-normalized sum by a smoothed sum, and then applies high-dimensional central limit theorem and Gaussian comparison tools to the sum of truncated vectors.","pith_inferences":["Editorial inference: if the main bound is correct, the one-dimensional case $d=1$ inherits a rate of only $n^{-1/8}$, far from the classical $n^{-1/2}$; checking the bound against the univariate self-normalized Berry–Esseen results would be a quick way to see how much slack the high-dimensional argument carries.","Editorial inference: the separation between $\\Delta_n$ and $\\Delta_n^X$ suggests that a bootstrap analogue, which the authors defer to future work, could be proved by recycling the same truncation and smoothing steps, yielding quantile-consistent confidence sets without estimating the full covariance matrix.","Editorial inference: the bound's dependence on $\\|E[Y_1]\\|_\\infty$ signals that asymmetric truncation bias is a main cost; a symmetrized truncation may reduce the $\\log$-exponent, though the paper does not pursue this."],"forward_implications":["If Theorem 1 is correct, the maximum of self-normalized sums has a valid Gaussian approximation in dimensions as large as $\\log d = o(n^{1/10})$, with no independence assumption across coordinates.","The error bound is explicit in both $n$ and $d$, so it can support finite-sample inference rather than only asymptotic statements.","Because the approximating Gaussian can be taken to have a correlation matrix with unit diagonal, conservative quantile corrections such as Bonferroni or Šidák remain available for simultaneous confidence statements.","The best-approximation version $\\Delta_n$ can vanish even when the second moment of $X_1$ does not exist, whereas the moment-matching version $\\Delta_n^X$ requires finite second moments."],"supporting_citations":[{"why":"Supplies the truncation construction for self-normalized sums and the scale $a_j$; its univariate Student-t Berry–Esseen method is the template being extended.","marker":"Bentkus and Götze (1996)"},{"why":"Provides the high-dimensional central limit theorem and Gaussian comparison inequalities used to control the sum-to-Gaussian and covariance-difference steps.","marker":"Chernozhuokov et al. (2022)"},{"why":"Supplies the smoothing lemma and derivative bounds for $H_{\\varepsilon,t}$ that control the Lindeberg-swap remainder.","marker":"Fang and Koike (2021)"},{"why":"Supplies the improved Gaussian comparison bound with a minimal-eigenvalue factor used in the sharper version of Lemma 5.","marker":"Lopes (2022)"},{"why":"Provides the smoothing lemma and the near-$n^{-1/2}$ high-dimensional central limit theorem benchmark the self-normalized result is compared with.","marker":"Kuchibhotla and Rinaldo (2020)"},{"why":"Establishes the earlier near-$n^{-\\kappa/2}$ rate for self-normalized sums under coordinate independence, the result this paper extends by dropping that assumption.","marker":"Das (2024)"},{"why":"Gives the explicit Gaussian anti-concentration bound, via Nazarov's inequality, used in the proof of the centered-sum comparison.","marker":"Chernozhukov et al. (2017)"}],"fun_headline_variants":["Explicit n^{-1/8} Berry-Esseen bound for high-dim self-normalized sums","High-dim self-normalized sums: n^{-1/8} Gaussian approximation bound","Berry-Esseen rate n^{-1/8} for maximum of self-normalized statistics","Self-normalized sums achieve n^{-1/8} Gaussian error in high dimensions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument leans on Lemma 5's factorization for the Gaussian covariance comparison and, in the corollary, on the expectation lower bound used to handle the case where the truncation level falls well below the variance; if either of those two steps fails, the stated rates are not established.","fun_headline_variants_meta":{"raw":{"variants":["Explicit n^{-1/8} Berry-Esseen bound for high-dim self-normalized sums","High-dim self-normalized sums: n^{-1/8} Gaussian approximation bound","Berry-Esseen rate n^{-1/8} for maximum of self-normalized statistics","Self-normalized sums achieve n^{-1/8} Gaussian error in high dimensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000278,"raw_usage":{"total_tokens":1698,"prompt_tokens":1032,"completion_tokens":666,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":648,"completion_tokens_details":{"reasoning_tokens":573}},"tokens_in":648,"tokens_out":666,"duration_ms":5769,"temperature":1.0,"reasoning_tokens":573,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:14:36.494103+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To test the key step, choose a centered distribution with a single coordinate pair whose truncation event is not independent of the product $e_j^\\top X_1\\,e_k^\\top X_1$, compute the true value $E[e_j^\\top Z_1^X e_k^\\top Z_1^X 1\\{E_{j,k}\\}] = E[e_j^\\top Z_1^X e_k^\\top Z_1^X]P(E_{j,k})$ and compare it with the paper's replacement $E[(e_j^\\top X_1e_k^\\top X_1/(n\\sigma_j\\sigma_k))1\\{E_{j,k}\\}]$; a numerical difference here means Lemma 5's covariance bound, and with it the stated rate, does not follow from the given proof.","supporting_citations":[{"cited_title":"and Koike, Y","cited_arxiv_id":null,"evidence_quote":"Supplies the smoothing lemma and derivative bounds for $H_{\\varepsilon,t}$ that control the Lindeberg-swap remainder."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the improved Gaussian comparison bound with a minimal-eigenvalue factor used in the sharper version of Lemma 5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the earlier near-$n^{-\\kappa/2}$ rate for self-normalized sums under coordinate independence, the result this paper extends by dropping that assumption."}],"review_version":1}