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REVIEW 2 major objections 4 minor 26 references

The Berry-Esseen Bound for High-dimensional Self-normalized Sums

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2501.08979 v1 pith:A5X2DKNN submitted 2025-01-15 math.PR math.STstat.TH

Woonyoung Chang , Kenta Takatsu , Konrad Urban , Arun Kumar Kuchibhotla This is my paper · ORCID
classification math.PRmath.STstat.TH MSC 60F0560G5062E20
keywords Berry-Esseenboundself-normalizedsumshigh-dimensionalcentrallimittheoremGaussianapproximationcoordinate-wisemaximumheavytailsKolmogorov-Smirnovdistancetruncatedmoments
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

2 major / 4 minor

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.

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 (2)
  1. [4, Proof of Corollary 1.1] 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.
  2. [4.1, Lemma 5, Eq. (15)] 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.
minor comments (4)
  1. [4.1, Lemma 3] The proof uses p for the dimension in the union bound, although the paper uses d throughout; p should be replaced by d.
  2. [4.1, Lemma 5] 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.
  3. [4.1, Lemma 6] 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].
  4. [4.2, Proposition 1, Eq. (31)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main bound is derived from external CLT, comparison, and smoothing results, and the self-citations are published, independent, and non-load-bearing.

full rationale

The derivation chain for Theorem 1 is self-contained in the sense relevant to circularity. The proof decomposes the Kolmogorov distance via a triangle inequality into truncation error (Lemma 3), self-normalization smoothing error (Lemma 4), the Gaussian comparison error (Lemma 5), and the centered-sum CLT error (inequalities (11) and (12)). Each piece is controlled by established external results: Bentkus-Götze's truncation lemma for the choice of a_j, Chernozhukov et al.'s smoothing and Gaussian comparison bounds, Lopes's comparison theorem, Nazarov's anti-concentration inequality, Fang-Koike's smoothing lemma, and Kuchibhotla-Patra's maximal inequalities. The self-citations (Kuchibhotla-Rinaldo, Kuchibhotla-Patra) are published results with stated assumptions that do not include the present paper's target bound; the smoothing lemma is also available from Fang-Koike, so no load-bearing weight falls on the self-citation. The truncation parameters a_j are deterministic functions of the law of X1, not parameters fitted to data or to the target quantity. No displayed equation in the paper reduces a claimed prediction to an input by construction, and no uniqueness claim is imported from the authors' prior work. The technical defects noted by the reader, such as the false normalization in the proof of Corollary 1.1 and the questionable Gaussian covariance factorization in Lemma 5, are mathematical correctness issues rather than instances of circular reasoning; they do not fit any of the enumerated circularity patterns and therefore do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No new entities are introduced; the Gaussian vectors are standard approximations. The proof leans on several imported theorems; the failure is not in the imports but in their application in Lemma 5 and Corollary 1.1.

assumptions (5)
  • standard math Truncation constants a_j exist and satisfy E[(e_j^T X1)^2 1{(e_j^T X1)^2 ≤ a_j^2 n}] = a_j^2 (Lemma 1.3 of Bentkus and Götze, 1996)
    Used to define truncated vectors Y_i with bounded sup-norm and coordinate-wise second moment 1/n; invoked in Section 2, Eq. (4)-(5).
  • standard math High-dimensional CLT for sums of independent vectors with matching second moments (Chernozhukov et al., 2022, and the paper's Proposition 1)
    Provides the Gaussian approximation for the sum Y, used in Eq. (12). The paper refines this theorem but relies on its Lemma A.1.
  • standard math Nazarov's Gaussian anti-concentration inequality (Chernozhukov et al., 2017)
    Controls the sliding window probability in the proof of Eq. (12).
  • standard math Decoupling inequality for U-statistics (de la Peña and Giné, 2012) and concentration bounds (Proposition B.1 of Kuchibhotla and Patra, 2022)
    Used in Lemmas 7 and 8 to bound moments of maxima of sums of truncated variables.
  • domain assumption Finite (2+δ)-th moment of the maximum standardized coordinate for Corollary 1.1
    Stated in Corollary 1.1 as the condition under which the rate log^{5/4}(ed)n^{-δ/8} is claimed.

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Cite this review

Pith. "Pith review of The Berry-Esseen Bound for High-dimensional Self-normalized Sums." pith.science (2026). https://pith.science/paper/A5X2DKNN

@misc{pith2026250108979,
  author       = {Pith},
  title        = {Pith review of: The Berry-Esseen Bound for High-dimensional Self-normalized Sums},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A5X2DKNN}},
  note         = {Machine review of arXiv:2501.08979}
}
abstract

This manuscript studies the Gaussian approximation of the coordinate-wise maximum of self-normalized statistics in high-dimensional settings. We derive an explicit Berry-Esseen bound under weak assumptions on the absolute moments. When the third absolute moment is finite, our bound scales as $\log^{5/4}(d)/n^{1/8}$ where $n$ is the sample size and $d$ is the dimension. Hence, our bound tends to zero as long as $\log(d)=o(n^{1/10})$. Our results on self-normalized statistics represent substantial advancements, as such a bound has not been previously available in the high-dimensional central limit theorem (CLT) literature.

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Reference graph

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