pith. sign in

arxiv: 2604.00642 · v2 · submitted 2026-04-01 · 🧮 math.PR

Quantitative central limit theorem for an integrated periodogram via the fourth moment theorem

Samir Ben Hariz , Duc-Quang Bui , Youssef Esstafa This is my paper

Pith reviewed 2026-05-13 22:18 UTC · model grok-4.3

classification 🧮 math.PR
keywords central limit theoremintegrated periodogramWasserstein distanceMalliavin-Stein methodfourth moment theoremlong memoryToeplitz quadratic formsregular variation
0
0 comments X

The pith

The integrated periodogram of a long-memory Gaussian sequence satisfies a quantitative central limit theorem in 1-Wasserstein distance.

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

The paper proves that the normalized integrated periodogram of a stationary Gaussian process converges to a standard normal distribution at an explicit rate in Wasserstein distance. The result holds under a regular-variation condition on the spectral density that permits singularities associated with long-range dependence and includes slowly varying corrections. The proof expresses the statistic as an element of the second Wiener chaos and invokes the fourth-moment theorem from the Malliavin-Stein method, which reduces normal approximation to verifying variance asymptotics together with an explicit upper bound on the fourth cumulant obtained from trace estimates on the associated integral operator. Kernel estimates, including Dirichlet-type bounds and a weighted Schur test, supply the necessary control over these traces. This supplies a quantitative strengthening of earlier qualitative central limit theorems for Toeplitz quadratic forms.

Core claim

Under a regular-variation assumption allowing long-memory singularities and slowly varying corrections, the integrated periodogram satisfies a quantitative central limit theorem in 1-Wasserstein distance. The argument represents the statistic in the second Wiener chaos and applies the Malliavin-Stein fourth-moment theorem, thereby reducing the normal approximation to variance asymptotics and an explicit control of the fourth cumulant via trace estimates for the associated integral operator.

What carries the argument

The fourth-moment theorem from the Malliavin-Stein method applied to the second Wiener chaos representation of the integrated periodogram, together with trace estimates on the associated integral operator obtained via Dirichlet-type kernel bounds and a weighted Schur test.

If this is right

  • The normalized integrated periodogram converges in distribution to a Gaussian limit at a rate controlled by the fourth cumulant.
  • Explicit variance asymptotics hold for the integrated periodogram when the spectral density satisfies the regular-variation condition.
  • The trace estimates supply uniform control on cumulants for processes whose spectral densities exhibit singularities at zero.
  • The same kernel estimates used for the integral operator extend the method to other quadratic forms of stationary Gaussian sequences.

Where Pith is reading between the lines

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

  • The quantitative bound could be used to construct asymptotic confidence intervals for spectral integrals in long-memory time series.
  • The kernel estimates developed for the integral operator may apply directly to other Toeplitz quadratic forms arising in nonparametric statistics.
  • Numerical evaluation of the Wasserstein distance for fractional Gaussian noise at specific Hurst indices would test whether the derived rate is sharp.

Load-bearing premise

The spectral density obeys a regular-variation condition that yields both the variance asymptotics and the trace estimates needed to bound the fourth cumulant.

What would settle it

A specific long-memory Gaussian sequence whose fourth cumulant fails to decay at the rate predicted by the trace estimates under the regular-variation condition would falsify the Wasserstein bound.

read the original abstract

We revisit the central limit theorem for integrated periodograms, equivalently for Toeplitz quadratic forms of stationary Gaussian sequences. Under a regular-variation assumption allowing long-memory singularities and slowly varying corrections, we prove a quantitative central limit theorem in 1-Wasserstein distance. The proof uses a second Wiener chaos representation and the Malliavin-Stein method (in particular, the Fourth Moment Theorem), reducing normal approximation to (i) variance asymptotics and (ii) an explicit control of the fourth cumulant via trace estimates for an associated integral operator. For convenience, we provide self-contained kernel estimates (Dirichlet-type bounds, convolution inequalities, and a weighted Schur test) used in the argument.

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 / 2 minor

Summary. The manuscript establishes a quantitative central limit theorem in 1-Wasserstein distance for the integrated periodogram of a stationary Gaussian sequence. Under a regular-variation assumption on the spectral density that accommodates long-memory singularities and slowly varying corrections, the integrated periodogram is represented as a second-chaos random variable; the Malliavin-Stein method (via the Fourth Moment Theorem) then reduces the normal approximation to explicit control of the variance asymptotics and the fourth cumulant through trace estimates on an associated integral operator, supported by self-contained kernel bounds (Dirichlet-type estimates, convolution inequalities, and a weighted Schur test).

Significance. If the stated bounds hold, the result supplies an explicit rate of convergence in Wasserstein distance for the CLT of integrated periodograms in the long-memory regime, extending earlier qualitative limit theorems. The reduction to variance asymptotics plus fourth-cumulant control via the Fourth Moment Theorem is direct and avoids additional parameters; the self-contained kernel estimates are a useful technical contribution for similar quadratic-form problems.

minor comments (2)
  1. [§2.2] §2.2, Assumption (A): the precise range of the regular-variation index (and the admissible slowly-varying functions) should be stated explicitly so that readers can immediately see which long-memory exponents are covered.
  2. [Theorem 3.1] The statement of the main theorem (Theorem 3.1) would benefit from a short remark clarifying whether the Wasserstein bound is uniform in the slowly-varying component or only asymptotic.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. The referee summary accurately captures the main results, including the reduction to variance asymptotics and fourth-cumulant bounds via the Fourth Moment Theorem, as well as the self-contained kernel estimates.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation reduces normal approximation of the integrated periodogram (second Wiener chaos) to two explicitly controlled quantities: variance asymptotics and fourth-cumulant bounds obtained from trace estimates on the associated integral operator. The paper supplies all required kernel estimates (Dirichlet-type bounds, convolution inequalities, weighted Schur test) self-contained under the stated regular-variation assumption on the spectral density. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear; the argument is internally consistent and uses only standard Malliavin-Stein machinery without reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard properties of Wiener chaos, the Malliavin-Stein fourth-moment theorem, and regular-variation assumptions on the spectral density; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Stationary Gaussian sequence admits a spectral representation with regularly varying spectral density
    Invoked to obtain variance asymptotics and to control the operator traces under long-memory singularities.
  • standard math Fourth-moment theorem applies to elements of the second Wiener chaos
    Standard result from Malliavin-Stein calculus used to reduce Wasserstein distance to fourth cumulant.

pith-pipeline@v0.9.0 · 5419 in / 1380 out tokens · 27116 ms · 2026-05-13T22:18:10.715105+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    Avram,On bilinear forms in gaussian random variables and Toeplitz matrices, Probability Theory and Related Fields79(1988), no

    F. Avram,On bilinear forms in gaussian random variables and Toeplitz matrices, Probability Theory and Related Fields79(1988), no. 1, 37–45. MR 0952991

    work page 1988

  2. [2]

    N. H. Bingham, C. M. Goldie, and J. L. Teugels,Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, 1987. MR 0898871

    work page 1987

  3. [3]

    P. J. Brockwell and R. A. Davis,Time series: Theory and methods, 2 ed., Springer Series in Statistics, Springer, 1991. MR 1093459

    work page 1991

  4. [4]

    Dahlhaus,Efficient parameter estimation for self-similar processes, Annals of Statistics17(1989), no

    R. Dahlhaus,Efficient parameter estimation for self-similar processes, Annals of Statistics17(1989), no. 4, 1749–1766. MR 1026311

    work page 1989

  5. [5]

    Fox and M

    R. Fox and M. S. Taqqu,Noncentral limit theorems for quadratic forms in random variables having long-range dependence, Annals of Probability13(1985), no. 2, 428–446. MR 0781415

    work page 1985

  6. [6]

    2, 517–532

    ,Large-sample properties of parameter estimates for strongly dependent stationary gaussian time se- ries, Annals of Statistics14(1986), no. 2, 517–532. MR 0840512

    work page 1986

  7. [7]

    2, 213–240

    ,Central limit theorems for quadratic forms in random variables having long-range dependence, Prob- ability Theory and Related Fields74(1987), no. 2, 213–240. MR 0871252

    work page 1987

  8. [8]

    M. S. Ginovyan,On Toeplitz type quadratic functionals of stationary gaussian processes, Probability Theory and Related Fields100(1994), 395–406. MR 1305588

    work page 1994

  9. [9]

    Ginovyan,Random Toeplitz functionals and their applications, Frontiers in Probability and the Statistical Sciences, Springer, Cham, [2025]©2025

    Mamikon S. Ginovyan,Random Toeplitz functionals and their applications, Frontiers in Probability and the Statistical Sciences, Springer, Cham, [2025]©2025. MR 4999369

    work page 2025

  10. [10]

    Ginovyan, Artur A

    Mamikon S. Ginovyan, Artur A. Sahakyan, and Murad S. Taqqu,The trace problem for Toeplitz matrices and operators and its impact in probability, Probab. Surv.11(2014), 393–440. MR 3290440

    work page 2014

  11. [11]

    Ginovyan and Murad S

    Mamikon S. Ginovyan and Murad S. Taqqu,Limit theorems for Toeplitz-type quadratic functionals of sta- tionary processes and applications, Probab. Surv.19(2022), 1–64. MR 4366223

    work page 2022

  12. [12]

    Giraitis and D

    L. Giraitis and D. Surgailis,A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittle’s estimate, Probability Theory and Related Fields86 (1990), 87–104. MR 1061950 14 SAMIR BEN HARIZ, DUC-QUANG BUI, AND YOUSSEF ESSTAF A

    work page 1990

  13. [13]

    Grenander and G

    U. Grenander and G. Szeg˝ o,Toeplitz forms and their applications, University of California Press, 1958. MR 0094840

    work page 1958

  14. [14]

    P. R. Halmos and V. S. Sunder,Bounded integral operators onL 2 spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 96, Springer, 1978. MR 0517709

    work page 1978

  15. [15]

    Lajos Horv´ ath and Qi-Man Shao,Limit theorems for quadratic forms with applications to Whittle’s estimate, Ann. Appl. Probab.9(1999), no. 1, 146–187. MR 1682588

    work page 1999

  16. [16]

    Hosoya,A limit theory for long-range dependence and statistical inference on related models, Annals of Statistics25(1997), no

    Y. Hosoya,A limit theory for long-range dependence and statistical inference on related models, Annals of Statistics25(1997), no. 1, 105–137. MR 1429919

    work page 1997

  17. [17]

    Katznelson,An introduction to harmonic analysis, 3 ed., Cambridge University Press, 2004

    Y. Katznelson,An introduction to harmonic analysis, 3 ed., Cambridge University Press, 2004. MR 2039503

    work page 2004

  18. [18]

    Nourdin and G

    I. Nourdin and G. Peccati,Stein’s method and exact Berry–Esseen asymptotics for functionals of gaussian fields, Annals of Probability37(2009), no. 6, 2231–2261. MR 2533469

    work page 2009

  19. [19]

    MR 2520122

    ,Stein’s method on Wiener chaos, Probability Theory and Related Fields145(2009), 75–118. MR 2520122

    work page 2009

  20. [20]

    MR 2962301

    ,Normal approximations with Malliavin calculus: From Stein’s method to universality, Cambridge Tracts in Mathematics, Cambridge University Press, 2012. MR 2962301

    work page 2012

  21. [21]

    Nualart,The malliavin calculus and related topics, 2 ed., Probability and its Applications, Springer, 2006

    D. Nualart,The malliavin calculus and related topics, 2 ed., Probability and its Applications, Springer, 2006. MR 2200233

    work page 2006

  22. [22]

    Simon,Trace ideals and their applications, 2 ed., Mathematical Surveys and Monographs, vol

    B. Simon,Trace ideals and their applications, 2 ed., Mathematical Surveys and Monographs, vol. 120, Amer- ican Mathematical Society, 2005. MR 2154153

    work page 2005

  23. [23]

    MR 1785484

    Masanobu Taniguchi and Yoshihide Kakizawa,Asymptotic theory of statistical inference for time series, Springer Series in Statistics, Springer-Verlag, New York, 2000. MR 1785484

    work page 2000

  24. [24]

    Zygmund,Trigonometric series, 3 ed., Cambridge Mathematical Library, Cambridge University Press,

    A. Zygmund,Trigonometric series, 3 ed., Cambridge Mathematical Library, Cambridge University Press,

    work page

  25. [25]

    MR 1963498 Le Mans Universit´e, LMM, Le Mans, France Email address:samir.ben hariz@univ-lemans.fr Le Mans Universit´e, LMM, Le Mans, France Email address:duc quang.bui@univ-lemans.fr Le Mans Universit´e, LMM, Le Mans, France Email address:youssef.esstafa@univ-lemans.fr

    work page