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Linear Spectral Statistics for Entrywise-Transformed Spiked Wigner Matrices under Shifted $L^4$ Profile Admissibility

T0 review · 0 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Rough entrywise transforms of spiked Wigner matrices still satisfy a Gaussian CLT for linear spectral statistics, with an explicit order-one mean.

desk verdict A careful, genuinely new LSS theorem for rough transforms of spiked Wigner matrices; the proof is coherent and the main caveat is the strength of W5, which the authors own. read the letter →

arxiv 2608.07820 v1 pith:M2IX4JSY submitted 2026-08-07 math.PR

Tsz-Kin Chan , Ji Oon Lee This is my paper · ORCID
classification math.PR MSC 60B2015B5260F05
keywords linearspectralstatisticsspikedWignermatricesentrywisenonlineartransformfour-momentreplacementshiftedprofileadmissibilityGaussianfluctuationgeneralizedoutliercontribution
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

The paper proves a central limit theorem for bulk linear spectral statistics of a Wigner matrix whose entries are first shifted by a rank-one spike and then passed through a possibly nonsmooth function f. The main claim is that, provided the shifted mean, variance, fourth cumulant, and fourth tails of f under the noise law satisfy five 'profile admissibility' conditions, the centered statistics converge jointly to Gaussian with the same zero-diagonal Wigner covariance that appears in the untransformed model, with fourth-cumulant parameter $\kappa_4^{f,\nu}$. The deterministic order-one centering is explicit: it separates the homogeneous Wigner bias, a rank-one Woodbury response, a zero-diagonal correction, and a quadratic variance-profile response. A separated supercritical outlier contributes exactly $\varphi(\theta+\theta^{-1})$ to the full-trace centering. A self-contained Gaussian corollary states that any centered, variance-normalized $f\in L^{4+\epsilon}(\gamma)$ satisfies the theorem with no pointwise differentiability of $f$.

What carries the argument

The load-bearing object is the three-point atomic completion (Proposition A.6): for any centered real variable $X$ with variance $v$ and finite fourth moment, there is an explicit three-point law $\sqrt{v}Q$ whose first four centered moments match $X$ exactly. At each edge shift $s_{ij}$, the paper applies this to $Y_s=f(\zeta+s)-\mathbb{E} f(\zeta+s)$, producing a uniformly bounded surrogate. The second key tool is a global Fourier–Duhamel estimate bounding the fifth derivative of $\mathrm{Tr}\tilde{\varphi}(A+N^{-1/2}uV_e)$ uniformly in $A$, which makes the edgewise replacement error summable once four moments match and the shifted fourth-moment Lindeberg condition W5 holds.

What would settle it

For Gaussian noise and $\varphi(t)=t^2$, the theorem predicts $\mathrm{Var}(\mathrm{Tr}(M_{N,\gamma}^f)^2)\to 4+2\kappa_f$ and $\mathbb{E}\mathrm{Tr}(M_{N,\gamma}^f)^2=N^{-1}+\lambda b_2+o(1)$. Compute these quantities exactly for a simple profile-admissible transform, for instance $f(t)=t$, with $\lambda>0$ and a delocalized spike; if either limit fails for large $N$, the covariance or centering formula is wrong.

Watch

Extended reading notes

Core claim

The central discovery is that an entrywise transform with no pointwise differentiability can be analyzed by matching only the first four centered moments at each microscopic shift. For every shift $s$, the paper constructs an explicit uniformly bounded three-point surrogate with the same mean, variance, and third and fourth centered moments as $f(\zeta+s)$, which reduces the rough transformed array to a bounded generalized-Wigner array without truncating $f$ or assuming coefficient stability. Applying the generalized-Wigner LSS theorem and a global Fourier–Duhamel fifth-derivative bound, the authors transfer the analytic linear statistic back to the raw transform and identify the limiting covariance as the standard zero-diagonal real-Wigner covariance with fourth-cumulant $\kappa_4^{f,\nu}$.

Load-bearing premise

The argument hinges on a uniform tail bound: after centering, the fourth moments of the shifted transformed entries, restricted to values above a large threshold, vanish uniformly over all small shifts as the threshold grows. If this uniformity fails, the bounded surrogate construction and the fifth-order replacement error summation both break down.

Editorial extensions

If this is right

  • The bulk LSS CLT holds without any derivative of the entry transform when W1–W5 hold; the diagonal-free real-Wigner covariance with $\kappa_4^{f,\nu}$ governs the fluctuations.
  • The order-one mean splits into four explicit channels, so the spike's effect on the mean is exactly computable from $a_1$, $b_2$, $\beta_2^{f,\nu}$, and the contour.
  • In the non-outlier regime $|\theta_{f,\nu}|\le 1$, the full trace equals the bulk statistic with probability tending to one, so the same CLT applies to the full trace.
  • In the supercritical regime $|\theta_{f,\nu}|>1$, a separated outlier contributes exactly $\varphi(\theta+\theta^{-1})$ to the full trace and no fluctuation at this scale.
  • For Gaussian noise, every centered, variance-normalized $f\in L^{4+\epsilon}(\gamma)$ is profile-admissible, yielding a derivative-free LSS CLT with explicit Hermite coefficients for all polynomial-growth transforms.

Reading between the lines

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

  • Beyond the paper, the same proof structure would likely extend if the uniform shifted-tail condition W5 were relaxed to a matrix-averaged condition along the actual edge shifts, since the bounded surrogate is needed only on the good set of edges.
  • Beyond the paper, extending the spike to finite rank should replace the scalar Woodbury denominator by an $r\times r$ determinant and add mixed quadratic variance-profile channels to the centering.
  • Beyond the paper, the explicit outlier contribution $\varphi(\theta+\theta^{-1})$ suggests that LSS-based detection tests could profitably separate bulk and outlier information in the supercritical regime.
  • Beyond the paper, replacing analytic test functions by Helffer–Sjöstrand representations could lower the required regularity of $\varphi$ independently of the roughness of the entry transform.
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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

0 major / 6 minor

Summary. The paper proves a bulk CLT for linear spectral statistics of entrywise-transformed rank-one spiked Wigner matrices with a general microscopic noise law. The entry transform f is only required to satisfy the shifted profile conditions W1–W5 and the spike is delocalized and weakly balanced. At each microscopic shift the authors construct an explicit bounded three-point variable with exactly the same first four centered moments, then apply a generalized-Wigner LSS theorem to the resulting bounded triangular array, compute the order-one deterministic mean as the sum of homogeneous, rank-one, zero-diagonal, and quadratic variance-profile responses, and transfer the CLT to the original rough transform via a global Fourier–Duhamel derivative estimate and a four-moment replacement telescope. The covariance is the standard zero-diagonal real-Wigner covariance with fourth-cumulant parameter κ4^{f,ν}. The paper also treats the supercritical outlier contribution to the full trace and gives a Gaussian-noise corollary for every centered, variance-normalized f in L^{4+ε}(γ), with explicit Hermite-type coefficients.

Significance. If the proof is correct, this is a substantial contribution to low-regularity LSS theory for transformed spiked Wigner matrices. The result removes differentiability assumptions on the entry transform, replaces them with transparent shifted-profile and fourth-tail conditions, and provides explicit formulas for all order-one centering terms and the fluctuation covariance. The three-point moment-matching construction, the unbalanced-profile Dyson computation, and the global Fourier–Duhamel replacement argument are all interesting in their own right. The Gaussian corollary is broad and readily usable in applications, covering every centered and variance-normalized polynomial-growth transform. The proof is internally coherent and, as far as I verified, has no circularity: the coefficients a1, b1, b2, and κ4^{f,ν} are fixed before the CLT is stated, and the imported Li–Xu theorem is applied only to an exactly stochastic, bounded, generalized-Wigner array. The flagged W5 condition is restrictive but is used precisely and explicitly in the two places where it is needed, namely the uniform bounded surrogate construction and the fifth-order replacement remainder estimate.

minor comments (6)
  1. [Lemma C.4, Eq. (223)] The inequality on the large-|u| region as printed, |Φ(u)-P(u)| ≤ C ε N^{-2}|u|^4, is not valid: for q=4 the term |Φ^{(4)}(0) u^4/4!| is only bounded by C N^{-2}|u|^4, and for very large |u| the Taylor remainder is controlled by C N^{-5/2}|u|^5. The argument still works if the right-hand side is replaced by C_ε N^{-2}|u|^4 with a constant depending on ε, since ε is fixed before the N→∞ limit and the Lindeberg condition sends the averaged tail to zero. I recommend correcting this display to avoid a false statement in the proof.
  2. [Appendix A and Section 2.1] The notation m(s) for the shifted mean E f(ζ+s) clashes with the Stieltjes transform m(z) used throughout the main text. Since both appear in the same proof chain, I suggest renaming the shifted mean, for instance μ(s) or m_f(s), to prevent confusion.
  3. [Proposition B.6] The proof relies on the global generalized-Wigner LSS theorem of Li–Xu [31, Theorem 2.2] without stating its hypotheses or the precise form of the characteristic-function expansion. Given that this theorem is a load-bearing input, the paper should either quote the relevant theorem explicitly or state its assumptions verbatim so that the verification in Proposition B.6 is checkable by the reader.
  4. [Proposition B.19] The proof of the hybrid-uniform exterior law passes from bounds on expectations of smooth test functions to probability estimates by using smooth approximations of indicators. A sentence explaining the uniformity over the replacement index r in this passage would improve readability, although the argument as written is sound.
  5. [Abstract and Introduction] The phrase 'derivative-free analytic linear spectral statistics theorem' might be misread as requiring no derivatives of the spectral test function. Since the theorem requires φ analytic, I suggest rewording to make explicit that the derivative-free statement concerns the entry transform f only.
  6. [Section 3.8] For the φ(t)=t^2 check, the statement that 'the rank-one contribution, including any separated outlier, is θ^2' is compressed. It would be clearer to display separately the bulk rank-one response, the zero-diagonal response, and the outlier contribution in the supercritical case, since the equality relies on the cancellation encoded in the contour convention.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the CLT is derived from explicit profile assumptions and an external generalized-Wigner backend, with no fitted or self-referential inputs.

full rationale

The paper's target quantities (a1, b1, b2, kappa_4^{f,nu}, theta_{f,nu}) are all defined directly from f and nu before any CLT is asserted; none is fitted to the data or to the linear spectral statistic being predicted. The proof reduces the rough entry array to a bounded three-point array matching the first four centered moments (Proposition A.6, Theorem A.8), then obtains a CLT for that array from the external generalized-Wigner LSS theorem of Li and Xu [31] (Proposition B.6, Lemma B.2), and finally transfers the result by a global Fourier-Duhamel fifth-derivative estimate (Lemma C.1, Lemma C.4, Theorem C.6). The centering m^{f,nu}_{phi,Gamma} and covariance V^Gamma_kappa are computed explicitly from the profile coefficients and contour integrals, and the low-degree checks in Section 3.8 confirm they are the correct deterministic constants rather than fitted values. The paper's self-citations ([13], [23], [30]) appear only in the literature review and are not used in any load-bearing proof step; no self-citation uniqueness theorem is invoked, and no known result is renamed as a new prediction. The restrictive uniform shifted fourth-moment condition W5 is a hypothesis, not a hidden input, and the paper itself proves in Proposition D.12 that bare L4(gamma) can fail W5, so the theorem is not a tautology. The external Li-Xu theorem is applied only to a bounded, exactly stochastic generalized-Wigner array, exactly the setting of that external theorem; this is independent support rather than circularity. No circular step could be identified in the derivation chain.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the new profile hypotheses W1-W5 and the spike conditions of Assumption 2.3, plus standard imported results used as black boxes. No free parameters are fitted: a1, b1, b2, kappa_4, and theta are functionals of f and nu. No new physical or mathematical entities are introduced.

assumptions (4)
  • domain assumption Condition 2.1 (W1-W5): fixed Borel transform f with E_nu f=0, E_nu f^2=1, shifted mean profile, shifted second-moment profile, shifted fourth-cumulant stability, and uniform shifted fourth-moment Lindeberg condition.
    These hypotheses define the admissible class of transforms and noise laws; they are not derived. The proof builds the three-point surrogate and the replacement argument on W3-W5. Invoked in Section 2.1 and throughout.
  • domain assumption Assumption 2.3: deterministic spike x satisfies ||x||_2=1, ||x||_infty=o(N^{-1/4}), and sum_i x_i = O(N^{o(1)}).
    Delocalization makes all microscopic shifts o(1), so a single uniform profile controls every entry; the weak balance condition suppresses the order-one linear variance-profile response. Invoked in Proposition A.4 and the Dyson reductions.
  • standard math Li-Xu generalized-Wigner LSS theorem [31, Theorem 2.2] including expectation and covariance formulas.
    Imported as the backend for the bounded atomic array in Proposition B.6; it supplies the CLT and the explicit bias and covariance kernels used to identify the endpoint.
  • standard math Generalized-Wigner isotropic local laws [10, 22, 2] and the triangular-array Bai-Yin theorem [37].
    Used in Propositions B.11, B.16, and B.19 for spectral confinement, the isotropic law, and the exterior law for the rough centered array.

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Pith. "Pith review of Linear Spectral Statistics for Entrywise-Transformed Spiked Wigner Matrices under Shifted $L^4$ Profile Admissibility." pith.science (2026). https://pith.science/paper/M2IX4JSY

@misc{pith2026260807820,
  author       = {Pith},
  title        = {Pith review of: Linear Spectral Statistics for Entrywise-Transformed Spiked Wigner Matrices under Shifted $L^4$ Profile Admissibility},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M2IX4JSY}},
  note         = {Machine review of arXiv:2608.07820}
}
abstract

We prove a derivative-free analytic linear spectral statistics theorem for entrywise-transformed rank-one spiked Wigner matrices with a general microscopic noise law. The transform is required to be centered, variance-normalized, and admissible under the small translations generated by the spike: its shifted mean and second moment have first- and second-order profiles, while its centered shifted fourth cumulants and fourth tails are stable. At every microscopic shift we construct an explicit uniformly bounded three-point variable matching the first four centered moments of the target entry exactly. A generalized-Wigner LSS theorem applies to the resulting bounded triangular array, and a global Fourier--Duhamel derivative estimate transfers the analytic statistic back to the rough transform without a common truncation, coefficient-stability assumption, or local law. The order-one mean consists of the homogeneous Wigner bias, a rank-one Woodbury response, a zero-diagonal correction, and a quadratic variance-profile response. The centered covariance is the standard zero-diagonal real-Wigner covariance with fourth-cumulant parameter $\kappa_4^{f,\nu}$. We distinguish the bulk contour statistic from the full trace in the supercritical regime and show that the separated outlier adds exactly $\varphi(\theta+\theta^{-1})$ to the full-trace centering. As a self-contained consequence, Gaussian noise with $f\in L^{4+\epsilon}(\gamma)$ satisfies the theorem without differentiability of $f$; the resulting bulk and full-trace corollary has explicit Hermite coefficients and includes every centered, variance-normalized polynomial-growth transform. Concrete likelihood-ratio, smooth-transform, bounded rough-transform, and atomic criteria are provided, together with obstructions showing that bare $L^4(\nu)$ is insufficient.

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Works this paper leans on

47 extracted references · 22 canonical work pages

  1. [1]

    Adhikari, I

    K. Adhikari, I. Jana, and K. Saha. Linear eigenvalue statistics of random matrices with a variance profile.Random Matrices: Theory and Applications, 10(3):2250004, 2021. doi:10.1142/S2010326322500046

    work page doi:10.1142/s2010326322500046 2021

  2. [2]

    O. H. Ajanki, L. Erd˝ os, and T. Kr¨ uger. Universality for general Wigner-type matrices. Probability Theory and Related Fields, 169(3–4):667–727, 2017. doi:10.1007/s00440-016- 0740-2

    work page doi:10.1007/s00440-016- 2017

  3. [3]

    O. H. Ajanki, L. Erd˝ os, and T. Kr¨ uger. Stability of the matrix Dyson equation and random matrices with correlations.Probability Theory and Related Fields, 173(1–2):293–373, 2019. doi:10.1007/s00440-018-0835-z

    doi:10.1007/s00440-018-0835-z 2019

  4. [4]

    Z. D. Bai and J. W. Silverstein. CLT for linear spectral statistics of large- dimensional sample covariance matrices.Annals of Probability, 32(1A):553–605, 2004. doi:10.1214/aop/1078415845

    arXiv 2004

  5. [5]

    Z. D. Bai and Y. Q. Yin. Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix.Annals of Probability, 16(4):1729–1741, 1988. doi:10.1214/aop/1176991594

    arXiv 1988

  6. [6]

    J. Baik, G. Ben Arous, and S. P´ ech´ e. Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices.Annals of Probability, 33(5):1643–1697, 2005. doi:10.1214/009117905000000233

    doi:10.1214/009117905000000233 2005

  7. [7]

    Baik and J

    J. Baik and J. W. Silverstein. Eigenvalues of large sample covariance matrices of spiked population models.Journal of Multivariate Analysis, 97(6):1382–1408, 2006. doi:10.1016/j.jmva.2005.08.003

    doi:10.1016/j.jmva.2005.08.003 2006

  8. [8]

    Benaych-Georges and A

    F. Benaych-Georges and A. Maltsev. Fluctuations of linear statistics of half-heavy-tailed random matrices.Stochastic Processes and their Applications, 126(11):3331–3352, 2016. doi:10.1016/j.spa.2016.04.030

    work page doi:10.1016/j.spa.2016.04.030 2016

Show all 47 references
  1. [9]

    Benaych-Georges and R

    F. Benaych-Georges and R. R. Nadakuditi. The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices.Advances in Mathematics, 227(1):494–521,

  2. [10]

    Bloemendal, L

    A. Bloemendal, L. Erd˝ os, A. Knowles, H.-T. Yau, and J. Yin. Isotropic local laws for sample covariance and generalized Wigner matrices.Electronic Journal of Probability, 19, article no. 33, 1–53, 2014. doi:10.1214/EJP.v19-3054

    2014 doi

  3. [11]

    Bloemendal and B

    A. Bloemendal and B. Vir´ ag. Limits of spiked random matrices I.Probability Theory and Related Fields, 156(3–4):795–825, 2013. doi:10.1007/s00440-012-0443-2

    2013 doi

  4. [12]

    Capitaine, C

    M. Capitaine, C. Donati-Martin, and D. F´ eral. The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations. Annals of Probability, 37(1):1–47, 2009. doi:10.1214/08-AOP394. 53

    2009 doi

  5. [13]

    H. W. Chung and J. O. Lee. Weak detection in the spiked Wigner model.IEEE Transactions on Information Theory, 68(11):7427–7453, 2022. doi:10.1109/TIT.2022.3185232

    2022

  6. [14]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, and D. Schr¨ oder. Optimal multi-resolvent local laws for Wigner matrices.Electronic Journal of Probability, 27, article no. 117, 1–38, 2022. doi:10.1214/22- EJP838

    2022 doi

  7. [15]

    Dallaporta and M

    S. Dallaporta and M. F´ evrier. Fluctuations of linear spectral statistics of deformed Wigner matrices. InS´ eminaire de Probabilit´ es LII, Lecture Notes in Mathematics, vol. 2363, pages 121–192, Springer, Cham, 2025. doi:10.1007/978-3-031-86422-3 3

    2025 doi

  8. [16]

    D. L. Donoho, M. Gavish, and I. M. Johnstone. Optimal shrinkage of eigenvalues in the spiked covariance model.Annals of Statistics, 46(4):1742–1778, 2018. doi:10.1214/17-AOS1601

    2018 doi

  9. [17]

    L. Erd˝ os. The matrix Dyson equation and its applications for random matrices. In A. Borodin, I. Corwin, and A. Guionnet, editors,Random Matrices, IAS/Park City Mathe- matics Series, vol. 26, pages 75–158. American Mathematical Society, Providence, RI, 2019. doi:10.1090/pcms/026/03

    2019 doi

  10. [18]

    Fan and W

    J. Fan and W. Wang. Asymptotics of empirical eigenstructure for high dimensional spiked covariance.Annals of Statistics, 45(3):1342–1374, 2017. doi:10.1214/16-AOS1487

    2017 doi

  11. [19]

    M. Feldman. Spectral properties of elementwise-transformed spiked matrices.SIAM Journal on Mathematics of Data Science, 7(2):542–571, 2025. doi:10.1137/23M1627985

    2025 doi

  12. [20]

    F´ eral and S

    D. F´ eral and S. P´ ech´ e. The largest eigenvalue of rank one deformation of large Wigner ma- trices.Communications in Mathematical Physics, 272(1):185–228, 2007. doi:10.1007/s00220- 007-0209-3

    2007 doi

  13. [21]

    Guionnet, J

    A. Guionnet, J. Ko, F. Krzakala, P. Mergny, and L. Zdeborov´ a. Spectral phase transitions in non-linear Wigner spiked models. arXiv:2310.14055, 2023. https://arxiv.org/abs/ 2310.14055

    2023 arXiv

  14. [22]

    Y. He, A. Knowles, and R. Rosenthal. Isotropic self-consistent equations for mean- field random matrices.Probability Theory and Related Fields, 171(1–2):203–249, 2018. doi:10.1007/s00440-017-0776-y

    2018 doi

  15. [23]

    H. C. Ji and J. O. Lee. Gaussian fluctuations for linear spectral statistics of de- formed Wigner matrices.Random Matrices: Theory and Applications, 9(3):2050011, 2020. doi:10.1142/S2010326320500112

    2020 doi

  16. [24]

    Johansson

    K. Johansson. On fluctuations of eigenvalues of random Hermitian matrices.Duke Mathe- matical Journal, 91(1):151–204, 1998. doi:10.1215/S0012-7094-98-09108-6

    1998 doi

  17. [25]

    I. M. Johnstone. On the distribution of the largest eigenvalue in principal components analysis.Annals of Statistics, 29(2):295–327, 2001. doi:10.1214/aos/1009210544

    2001

  18. [26]

    J. H. Jung, H. W. Chung, and J. O. Lee. Detection of signal in the spiked rectangular models. InProceedings of the 38th International Conference on Machine Learning, PMLR 139:5158–5167, 2021.https://proceedings.mlr.press/v139/jung21a.html

    2021

  19. [27]

    Knowles and J

    A. Knowles and J. Yin. The isotropic semicircle law and deformation of Wigner ma- trices.Communications on Pure and Applied Mathematics, 66(11):1663–1749, 2013. doi:10.1002/cpa.21450

    2013 doi

  20. [28]

    Landon and P

    B. Landon and P. Sosoe. Almost-optimal bulk regularity conditions in the CLT for Wigner matrices. arXiv:2204.03419, 2022.https://arxiv.org/abs/2204.03419. 54

    2022 arXiv

  21. [29]

    R. Lata la. Some estimates of norms of random matrices.Proceedings of the American Mathematical Society, 133(5):1273–1282, 2005. doi:10.1090/S0002-9939-04-07800-1

    2005 doi

  22. [30]

    Lee and J

    A. Lee and J. O. Lee. Fluctuations of the largest eigenvalues of transformed spiked Wigner matrices. InProceedings of the 42nd International Conference on Machine Learning, Proceedings of Machine Learning Research, 267:33322–33351, 2025. https://proceedings. mlr.press/v267/lee25t.html

    2025

  23. [31]

    Li and Y

    Y. Li and Y. Xu. On fluctuations of global and mesoscopic linear statistics of generalized Wigner matrices.Bernoulli, 27(2):1057–1076, 2021. doi:10.3150/20-BEJ1265

    2021 doi

  24. [32]

    Lodhia and A

    A. Lodhia and A. Maltsev. Covariance kernel of linear spectral statistics for half-heavy tailed Wigner matrices.Random Matrices: Theory and Applications, 12(2):2250054, 2023. doi:10.1142/S201032632250054X

    2023 doi

  25. [33]

    Lytova and L

    A. Lytova and L. Pastur. Central limit theorem for linear eigenvalue statistics of ran- dom matrices with independent entries.Annals of Probability, 37(5):1778–1840, 2009. doi:10.1214/09-AOP452

    2009 doi

  26. [34]

    Moniri and H

    B. Moniri and H. Hassani. Signal-plus-noise decomposition of nonlinear spiked random matrix models. arXiv:2405.18274, 2024.https://arxiv.org/abs/2405.18274

    2024 arXiv

  27. [35]

    B. Nadler. Finite sample approximation results for principal component analysis: A matrix perturbation approach.Annals of Statistics, 36(6):2791–2817, 2008. doi:10.1214/08-AOS618

    2008 doi

  28. [36]

    A. Onatski. Asymptotics of the principal components estimator of large factor mod- els with weakly influential factors.Journal of Econometrics, 168(2):244–258, 2012. doi:10.1016/j.jeconom.2012.01.034

    2012 doi

  29. [37]

    O’Rourke, D

    S. O’Rourke, D. Renfrew, and A. Soshnikov. On fluctuations of matrix entries of regular functions of Wigner matrices with non-identically distributed entries.Journal of Theoretical Probability, 26(3):750–780, 2013. doi:10.1007/s10959-011-0396-x

    2013 doi

  30. [38]

    Passemier, M

    D. Passemier, M. R. McKay, and Y. Chen. Asymptotic linear spectral statistics for spiked Hermitian random matrices.Journal of Statistical Physics, 160(1):120–150, 2015. doi:10.1007/s10955-015-1233-x

    2015 doi

  31. [39]

    D. Paul. Asymptotics of sample eigenstructure for a large dimensional spiked covariance model.Statistica Sinica, 17(4):1617–1642, 2007. https://www3.stat.sinica.edu.tw/ statistica/j17n4/j17n418/j17n418.html

    2007

  32. [40]

    Perry, A

    A. Perry, A. S. Wein, A. S. Bandeira, and A. Moitra. Optimality and sub-optimality of PCA I: spiked random matrix models.Annals of Statistics, 46(5):2416–2451, 2018. doi:10.1214/17-AOS1625

    2018 doi

  33. [41]

    Pizzo, D

    A. Pizzo, D. Renfrew, and A. Soshnikov. On finite rank deformations of Wigner matri- ces.Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques, 49(1):64–94, 2013. doi:10.1214/11-AIHP459

    2013 doi

  34. [42]

    V. Riabov. Mesoscopic eigenvalue statistics for Wigner-type matrices.Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques, 61(1):129–154, 2025. doi:10.1214/23-AIHP1438

    2025 doi

  35. [43]

    Shcherbina

    M. Shcherbina. Central limit theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices.Journal of Mathematical Physics, Analysis, Ge- ometry, 7(2):176–192, 2011. https://jmag.ilt.kharkiv.ua/index.php/jmag/article/ view/jm07-0176e. 55

    2011

  36. [44]

    Sosoe and P

    P. Sosoe and P. Wong. Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices.Advances in Mathematics, 249:37–87, 2013. doi:10.1016/j.aim.2013.09.004

    2013 doi

  37. [45]

    V. H. Vu. Spectral norm of random matrices.Combinatorica, 27(6):721–736, 2007. doi:10.1007/s00493-007-2190-z

    2007 doi

  38. [46]

    Q. Wang, J. W. Silverstein, and J.-F. Yao. A note on the CLT of the LSS for sample covariance matrix from a spiked population model.Journal of Multivariate Analysis, 130:194–207, 2014. doi:10.1016/j.jmva.2014.04.021. 56

    2014 doi

  39. [2011]

    doi:10.1016/j.aim.2011.02.007

    2011 doi

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