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arxiv: 1906.09063 · v1 · pith:4SX5UIGCnew · submitted 2019-06-21 · 🧮 math.PR

Normal Approximation for Weighted Sums under a Second Order Correlation Condition

S. G. Bobkov , G.P. Chistyakov , F. G\"otze This is my paper

Pith reviewed 2026-05-25 18:51 UTC · model grok-4.3

classification 🧮 math.PR MSC 60F0560E15
keywords normal approximationKolmogorov distanceweighted sumsdependent variablescorrelation conditionconcentration inequalitieslog-concave measures
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The pith

Under a second-order correlation condition, weighted sums of dependent summands approximate the normal law with average Kolmogorov distance of order (log n)/n.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a convergence rate in the central limit theorem for weighted sums where the summands are dependent. It shows that a second-order correlation condition on the variables produces an upper bound of order (log n)/n on the average Kolmogorov distance to the normal distribution. The argument rests on sharpened concentration inequalities for functions on high-dimensional spheres. The bound is illustrated for the case of log-concave measures. A reader would care because the rate improves on many existing bounds that require full independence or stronger mixing assumptions.

Core claim

Under correlation-type conditions, an upper bound of order (log n)/n holds for the average Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law; the result follows from improved concentration inequalities on high-dimensional Euclidean spheres and is illustrated on log-concave probability measures.

What carries the argument

The second-order correlation condition, which limits pairwise dependence among the summands sufficiently to obtain the stated rate via sphere concentration.

If this is right

  • The rate applies directly to weighted sums arising from log-concave probability measures.
  • The same sphere-concentration method yields quantitative bounds whenever the second-order condition can be verified.
  • The average distance controls the typical rather than worst-case deviation from normality across choices of weights.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may extend to other dependence structures that admit similar sphere-concentration estimates, such as certain graphical models.
  • The (log n)/n rate suggests that moderate dependence does not degrade the normal approximation beyond a logarithmic factor in high dimensions.
  • Numerical checks on finite samples from log-concave distributions could test whether the bound is observed in practice.

Load-bearing premise

The second-order correlation condition on the summands holds and the improved concentration inequalities on high-dimensional Euclidean spheres are valid.

What would settle it

A concrete family of dependent variables satisfying all other hypotheses but violating the second-order correlation condition for which the average Kolmogorov distance remains larger than C(log n)/n for some constant C and infinitely many n.

read the original abstract

Under correlation-type conditions, we derive an upper bound of order $(\log n)/n$ for the average Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. The result is based on improved concentration inequalities on high-dimensional Euclidean spheres. Applications are illustrated on the example of log-concave probability measures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims that under a second-order correlation condition on dependent summands, the average Kolmogorov distance between the law of a weighted sum and the standard normal is bounded above by a quantity of order (log n)/n. The argument proceeds by deriving improved concentration inequalities on high-dimensional Euclidean spheres and applying them to obtain the normal approximation bound; an application to log-concave measures is sketched.

Significance. If the derivation is valid, the result supplies an explicit rate for normal approximation of weighted sums under a correlation-type hypothesis that is weaker than independence yet still yields a near-optimal bound. The use of sphere-concentration tools is a methodological strength that may extend to other geometric probability settings. The paper ships an explicit, non-asymptotic bound rather than an existence statement.

minor comments (3)
  1. The precise definition of the 'average' Kolmogorov distance (over which measure on the weights?) should be stated explicitly in the main theorem rather than left implicit from the abstract.
  2. The statement of the second-order correlation condition would benefit from an immediate comparison with the classical pairwise-correlation or covariance-decay assumptions used in earlier Berry–Esseen-type results for dependent variables.
  3. In the application section, the verification that log-concave measures satisfy the second-order condition should include a short calculation or reference to the relevant property of the measure.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report lists no specific major comments, so we have no point-by-point responses. We are prepared to incorporate any minor suggestions if provided separately.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives an O((log n)/n) bound on average Kolmogorov distance for weighted sums under an explicitly stated second-order correlation condition, relying on concentration inequalities for high-dimensional spheres and applications to log-concave measures. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the argument applies stated hypotheses to obtain the bound without the target result being presupposed in the inputs or definitions. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities listed. The result rests on unspecified correlation conditions and sphere concentration inequalities whose validity is assumed.

pith-pipeline@v0.9.0 · 5576 in / 1040 out tokens · 15062 ms · 2026-05-25T18:51:17.856352+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    Anttila, K

    M. Anttila, K. Ball, and I. Perissinaki. The central limit problem for convex bodies. Trans. Amer. Math. Soc. 355 (2003), no. 12, 4723--4735

    work page 2003

  2. [2]

    S. G. Bobkov. On concentration of distributions of random weighted sums. Ann. Probab. 31 (2003), no. 1, 195--215

    work page 2003

  3. [3]

    S. G. Bobkov. Concentration of distributions of the weighted sums with Bernoullian coefficients. Geometric aspects of functional analysis, 27--36, Lecture Notes in Math., 1807, Springer, Berlin, 2003

    work page 2003

  4. [4]

    S. G. Bobkov. Concentration of normalized sums and a central limit theorem for noncorrelated random variables. Ann. Probab. 32 (2004), no. 4, 2884--2907

    work page 2004

  5. [5]

    S. G. Bobkov. On a theorem of V. N. Sudakov on typical distributions. (Russian) J. Math. Sciences (New York), 167 (2010), no. 4, 464--473. Translated from: Zap. Nauchn. Semin. POMI, vol. 368 (2009), 59--74

    work page 2010

  6. [6]

    S. G. Bobkov, G. P. Chistyakov, and F. G\"otze. Second-order concentration on the sphere. Commun. Contemp. Math. 19 (2017), no. 5, 1650058, 20 pp

    work page 2017

  7. [7]

    S. G. Bobkov, G. P. Chistyakov, and F. G\"otze. Gaussian mixtures and normal approximation for V. N. Sudakov's typical distributions. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 457 (2017), Veroyatnosti Statistika. 25, 37--52

    work page 2017

  8. [8]

    S. G. Bobkov, G. P. Chistyakov, and F. G\"otze. Berry-Esseen bounds for typical weighted sums. J. Electron. Probab. 23 (2018), no. 92, 1--22

    work page 2018

  9. [9]

    S. G. Bobkov, G. P. Chistyakov, and F. G\"otze. Concentration and Gaussian approximation for randomized sums. In preparation

    work page

  10. [10]

    S. G. Bobkov, and F. G\"otze. Concentration inequalities and limit theorems for randomized sums. Probab. Theory Related Fields 137 (2007), no. 1--2, 49--81

    work page 2007

  11. [11]

    S. G. Bobkov, and A. Koldobsky. On the central limit property of convex bodies. Geometric aspects of functional analysis, 44--52, Lecture Notes in Math., 1807, Springer, Berlin, 2003

    work page 2003

  12. [12]

    C. Borell. Convex measures on locally convex spaces. Ark. Mat. 12 (1974), 239--252

    work page 1974

  13. [13]

    Brazitikos, A

    S. Brazitikos, A. Giannopoulos, P. Valettas, and B.-H. Vritsiou. Geometry of isotropic convex bodies. Mathematical Surveys and Monographs, 196. American Mathematical Society, Providence, RI, 2014. xx+594 pp

    work page 2014

  14. [14]

    Diaconis, and D

    P. Diaconis, and D. Freedman. Asymptotics of graphical projection pursuit. Ann. Stat. 12 (1984), no. 3, 793--815

    work page 1984

  15. [15]

    R. Eldan. Thin shell implies spectral gap up to polylog via a stochastic localization scheme. Geom. Funct. Anal. 23 (2013), no. 2, 532--569

    work page 2013

  16. [16]

    Eldan, and B

    R. Eldan, and B. Klartag. Pointwise estimates for marginals of convex bodies. J. Funct. Anal. 254 (2008), no. 8, 2275--2293

    work page 2008

  17. [17]

    Gu\'edon, and E

    O. Gu\'edon, and E. Milman. Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures. Geom. Funct. Anal. 21 (2011), no. 5, 1043--1068

    work page 2011

  18. [18]

    B. Klartag. A central limit theorem for convex sets. Invent. Math. 168 (2007), no. 1, 91--131

    work page 2007

  19. [19]

    B. Klartag. Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal. 245 (2007), no. 1, 284--310

    work page 2007

  20. [20]

    B. Klartag. A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields 145 (2009), no. 1-2, 1--33

    work page 2009

  21. [21]

    Klartag, and S

    B. Klartag, and S. Sodin. Variations on the Berry-Esseen theorem. Teor. Veroyatn. Primen. 56 (2011), no. 3, 514--533; reprinted in: Theory Probab. Appl. 56 (2012), no. 3, 403--419

    work page 2011

  22. [22]

    M. Ledoux. Concentration of measure and logarithmic Sobolev inequalities. S\'eminaire de Probabilit\'es XXXIII. Lect. Notes in Math. 1709 (1999), 120--216, Springer

    work page 1999

  23. [23]

    Y. T. Lee, and S. S. Vempala. Eldan's stochastic localization and the KLS hyperplane conjecture: an improved lower bound for expansion. 58th Annual IEEE Symposium on Foundations of Computer Science -- FOCS 2017, 998--1007, IEEE Computer Soc., Los Alamitos, CA, 2017

    work page 2017

  24. [24]

    M. W. Meckes. Gaussian marginals of convex bodies with symmetries. Beitr\"age Algebra Geom. 50 (2009), no. 1, 101--118

    work page 2009

  25. [25]

    E. S. Meckes, and M. W. Meckes. The central limit problem for random vectors with symmetries. J. Theoret. Probab. 20 (2007), no. 4, 697--720

    work page 2007

  26. [26]

    V. D. Milman, and G. Schechtman. Asymptotic theory of finite-dimensional normed spaces. With an appendix by M. Gromov. Lecture Notes in Mathematics, 1200. Springer-Verlag, Berlin, 1986. viii+156 pp. 2mm

    work page 1986

  27. [27]

    S. V. Nagaev. On the distribution of linear functionals in finite-dimensional spaces of large dimension. (Russian) Dokl. Akad. Nauk SSSR 263 (1982), no. 2, 295--297

    work page 1982

  28. [28]

    V. V. Petrov. Sums of independent random variables. Translated from the Russian by A. A. Brown. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. Springer--Verlag, New York--Heidelberg, 1975. x+346 pp. 2mm

    work page 1975

  29. [29]

    V. V. Petrov. Limit theorems for sums of independent random variables (Russian), Nauka, Moscow, 1987. 318 pp

    work page 1987

  30. [30]

    V. N. Sudakov. Typical distributions of linear functionals in finite-dimensional spaces of high dimension. (Russian) Soviet Math. Dokl. 19 (1978), 1578--1582; translation in: Dokl. Akad. Nauk SSSR, 243 (1978), no. 6, 1402--1405

    work page 1978

  31. [31]

    von Weizs\"acker

    H. von Weizs\"acker. Sudakov's typical marginals, random linear functionals and a conditional central limit theorem. Probab. Theory Rel. Fields, 107 (1997), 313--324

    work page 1997

  32. [32]

    S. G. Bobkov, and A. Koldobsky. On the central limit property of convex bodies. Geometric aspects of functional analysis, 44--52, Lecture Notes in Math. 1807, Springer, Berlin, 2003. 2mm

    work page 2003

  33. [33]

    Brehm, and J

    U. Brehm, and J. Voigt. Asymptotics of cross sections for convex bodies. Beitr\"age Algebra Geom. 41 (2000), no. 2, 437--454. 2mm

    work page 2000

  34. [34]

    S. Sodin. Tail-sensitive Gaussian asymptotics for marginals of concentrated measures in high dimension. Geometric aspects of functional analysis, 271--295, Lecture Notes in Math. 1910, Springer, Berlin, 2007

    work page 1910