Geodesic
Sufficient conditions for the Marchenko–Pastur theorem
Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 4, pp. 813-833 Cet article a éte moissonné depuis la source Math-Net.Ru

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We find general sufficient conditions in the Marchenko–Pastur theorem for high-dimensional sample covariance matrices associated with random vectors, for which the weak concentration property of quadratic forms may not hold in general. The results are obtained by means of a new martingale method, which may be useful in other problems of random matrix theory.
Mots-clés : random matrices
Keywords: sample covariance matrices, the Marchenko–Pastur law. 
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P. A. Yaskov. Sufficient conditions for the Marchenko–Pastur theorem. Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 4, pp. 813-833. http://geodesic.mathdoc.fr/item/TVP_2023_68_4_a8/

[1] R. Adamczak, “On the Marchenko–Pastur and circular laws for some classes of random matrices with dependent entries”, Electron. J. Probab., 16 (2011), 37, 1065–1095 | DOI | MR | Zbl

[2] R. Adamczak, “Some remarks on the Dozier–Silverstein theorem for random matrices with dependent entries”, Random Matrices Theory Appl., 2:2 (2013), 1250017, 46 pp. | DOI | MR | Zbl

[3] S. Anatolyev, P. Yaskov, “Asymptotics of diagonal elements of projection matrices under many instruments/regressors”, Econometric Theory, 33:3 (2017), 717–738 | DOI | MR | Zbl

[4] Zhidong Bai, Wang Zhou, “Large sample covariance matrices without independence structures in columns”, Statist. Sinica, 18:2 (2008), 425–442 | MR | Zbl

[5] Zhidong Bai, J. W. Silverstein, Spectral analysis of large dimensional random matrices, Springer Ser. Statist., 2nd ed., Springer, New York, 2010, xvi+551 pp. | DOI | MR | Zbl

[6] M. Banna, F. Merlevède, M. Peligrad, “On the limiting spectral distribution for a large class of symmetric random matrices with correlated entries”, Stochastic Process. Appl., 125:7 (2015), 2700–2726 | DOI | MR | Zbl

[7] J. D. Batson, D. A. Spielman, N. Srivastava, “Twice-Ramanujan sparsifiers”, STOC {'09}: Proceedings the 41st annual ACM symposium on theory of computing, ACM, New York, 2009, 255–262 | DOI | MR | Zbl

[8] J. Bryson, R. Vershynin, Hongkai Zhao, “Marchenko–Pastur law with relaxed independence conditions”, Random Matrices Theory Appl., 10:4 (2021), 2150040, 28 pp. | DOI | MR

[9] D. Chafaï, K. Tikhomirov, “On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence”, Probab. Theory Related Fields, 170:3-4 (2018), 847–889 | DOI | MR | Zbl

[10] B. Collins, Jianfeng Yao, Wangjun Yuan, “On spectral distribution of sample covariance matrices from large dimensional and large $k$-fold tensor products”, Electron. J. Probab., 27 (2022), 102, 18 pp. | DOI | MR | Zbl

[11] V. L. Girko, A. K. Gupta, “Asymptotic behavior of spectral function of empirical covariance matrices”, Random Oper. Stochastic Equations, 2:1 (1994), 43–60 | DOI | MR | Zbl

[12] F. Götze, A. A. Naumov, A. N. Tikhomirov, “Limit theorems for two classes of random matrices with dependent entries”, Theory Probab. Appl., 59:1 (2015), 23–39 | DOI | DOI | MR | Zbl

[13] A. Lytova, “Central limit theorem for linear eigenvalue statistics for a tensor product version of sample covariance matrices”, J. Theoret. Probab., 31:2 (2018), 1024–1057 | DOI | MR | Zbl

[14] V. A. Marčenko, L. A. Pastur, “Distribution of eigenvalues for some sets of random matrices”, Math. USSR-Sb., 1:4 (1967), 457–483 | DOI | MR | Zbl

[15] A. Pajor, L. Pastur, “On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution”, Studia Math., 195:1 (2009), 11–29 | DOI | MR | Zbl

[16] L. Pastur, M. Shcherbina, Eigenvalue distribution of large random matrices, Math. Surveys Monogr., 171, Amer. Math. Soc., Providence, RI, 2011, xiv+632 pp. | DOI | MR | Zbl

[17] N. Srivastava, R. Vershynin, “Covariance estimation for distributions with $2+\varepsilon$ moments”, Ann. Probab., 41:5 (2013), 3081–3111 | DOI | MR | Zbl

[18] P. Yaskov, “Lower bounds on the smallest eigenvalue of a sample covariance matrix”, Electron. Commun. Probab., 19 (2014), 83, 10 pp. | DOI | MR | Zbl

[19] P. Yaskov, “Sharp lower bounds on the least singular value of a random matrix without the fourth moment condition”, Electron. Commun. Probab., 20 (2015), 44, 9 pp. | DOI | MR | Zbl

[20] P. Yaskov, “Variance inequalities for quadratic forms with applications”, Math. Methods Statist., 24:4 (2015), 309–319 | DOI | MR | Zbl

[21] P. Yaskov, “A short proof of the Marchenko–Pastur theorem”, C. R. Math. Acad. Sci. Paris, 354:3 (2016), 319–322 | DOI | MR | Zbl

[22] P. Yaskov, “Necessary and sufficient conditions for the Marchenko–Pastur theorem”, Electron. Commun. Probab., 21 (2016), 73, 8 pp. | DOI | MR | Zbl

[23] P. Yaskov, “LLN for quadratic forms of long memory time series and its applications in random matrix theory”, J. Theoret. Probab., 31:4 (2018), 2032–2055 | DOI | MR | Zbl

[24] P. Yaskov, “Limit of the smallest eigenvalue of a sample covariance matrix for spherical and related distributions”, Recent developments in stochastic methods and applications, ICSM-5 (Moscow, 2020), Springer Proc. Math. Statist., 371, Springer, Cham, 2021, 229–241 | DOI | MR

[25] P. A. Yaskov, “Limiting spectral distribution for large sample covariance matrices with graph-dependent elements”, Theory Probab. Appl., 67:3 (2022), 375–388 | DOI | DOI | MR | Zbl

[26] P. Yaskov, “Marchenko–Pastur law for a random tensor model”, Electron. Commun. Probab., 28 (2023), 23, 17 pp. | DOI | MR | Zbl

[27] Y. Q. Yin,, Z. D. Bai, P. R. Krishnaiah, “On the limit of the largest eigenvalue of the large dimensional sample covariance matrix”, Probab. Theory Related Fields, 78:4 (1988), 509–521 | DOI | MR | Zbl