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Random weighted projections, random quadratic forms and random eigenvectors. (English) Zbl 1384.60029

Summary: We present a concentration result concerning random weighted projections in high dimensional spaces. As applications, we prove (1) New concentration inequalities for random quadratic forms. (2) The infinity norm of most unit eigenvectors of a random \(\pm 1\) matrix is of order \(O(\sqrt{\log n/n})\). (3) An estimate on the threshold for the local semi-circle law which is tight up to a \(\sqrt{\log n}\) factor.

MSC:

60B20 Random matrices (probabilistic aspects)
60E15 Inequalities; stochastic orderings
47B80 Random linear operators
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[1] Arous, Extreme gaps between eigenvalues of random matrices, Ann Probab 41 pp 2648– (2013) · Zbl 1282.60008
[2] Bai, Convergence rates of the spectral distributions of large Wigner matrices, Int Math J 1 pp 65– (2002) · Zbl 0987.60050
[3] Bai, On the empirical distribution of eigenvalues of a class of large dimensional random matrices, J Multivar Anal 54 pp 175– (1995) · Zbl 0833.60038
[4] Bai, Spectral analysis of large dimensional random matrices, Spring Series in Statistics (2010) · Zbl 1301.60002
[5] Bai, Limit of the smallest eigenvalue of a large dimensional sample covariance matrix, Ann Probab 21 pp 1275– (1993) · Zbl 0779.60026
[6] C. Cacciapuoti A. Maltsev B. Schlein Local Marchenko-Pastur law at the hard edge of sample covariance matrices 2012 · Zbl 1282.15031
[7] Dekel, Eigenvectors of random graphs: Nodal domains, Random Struct Algorithms 39 pp 39– (2011) · Zbl 1223.05275
[8] Erdos, Universality of Wigner random matrices: a survey of recent results, Russ Math Surv 66 pp 507– (2011) · Zbl 1230.82032
[9] Erdos, Local semicircle law and complete delocalization for Wigner random matrices, Commun Math Phys 287 pp 641– (2009) · Zbl 1186.60005
[10] Erdos, Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices, Ann Probab 37 pp 815– (2009) · Zbl 1175.15028
[11] Erdos, Wegner estimate and level repulsion for Wigner random matrices, Int Math Res Notices 2010 pp 436– (2010)
[12] Erdos, The local relaxation flow approach to universality of the local statistics for random matrices, Ann Int H Poincaré Probab Stat 48 pp 1– (2012) · Zbl 1285.82029
[13] Erdos, Bulk universality for generalized Wigner matrices, Probab Theory Relat Fields 154 pp 341– (2012) · Zbl 1277.15026
[14] Hanson, A bound on tail probabilities for quadratic forms in independent random variables, Ann Math Statist 42 pp 1079– (1971) · Zbl 0216.22203
[15] Geman, A limit theorem for the norm of random matrices, Ann Probab 8 pp 252– (1980) · Zbl 0428.60039
[16] Hsu, A tail inequality for quadratic forms of subgaussian random vectors, Electron Commun Probab 17 pp 1– (2012) · Zbl 1309.60017
[17] Ledoux, Mathematical Surveys and Monographs, 89 (2001)
[18] Marčenko, Distribution of eigenvalues for some sets of random matrices, Math USSR-Sbornik 1 pp 457– (1967) · Zbl 0162.22501
[19] Mehta, Random matrices, Pure and Applied Mathematics (Amsterdam), 142 (2004) · Zbl 1107.15019
[20] Nguyen, Random matrices: Law of the determinant, Ann Probab 42 pp 146– (2014) · Zbl 1299.60005
[21] Pastur, On the spectrum of random matrices, Theor Math Phys 10 pp 67– (1972) · Zbl 0252.35051
[22] N.S. Pillai J. Yin Universality of covariance matrices 2011
[23] M. Rudelson R. Vershynin Hanson-Wright inequality and sub-gaussian concentration 2013 · Zbl 1329.60056
[24] M. Rudelson R. Vershynin Delocalization of eigenvectors of random matrices with independent entries 2013
[25] Samson, Concentration of measure inequalities for Markov chains and {\(\Phi\)}-mixing processes, Ann Probab 28 pp 416– (2000) · Zbl 1044.60061
[26] Tao, On random {\(\pm\)}1 matrices: singularity and determinant, Random Struct Algorithms 28 pp 1– (2006) · Zbl 1086.60008
[27] Tao, Random matrices: The distribution of the smallest singular values, Geom Funct Anal 20 pp 260– (2010) · Zbl 1210.60014
[28] Tao, Random matrices: Universality of local eigenvalue statistics up to the edge, Commun Math Phys 298 pp 549– (2010) · Zbl 1202.15038
[29] Tao, Random matrices: Universality of local eigenvalue statistics, Acta Math 206 pp 127– (2011) · Zbl 1217.15043
[30] Tao, Random covariance matrices: Universality of local statistics of eigenvalues, Ann Probab 40 pp 1285– (2012) · Zbl 1247.15036
[31] T. Tao V. Vu Random matrices: The universality phenomenon for Wigner ensembles 2012
[32] Tran, Sparse random graphs: eigenvalues and eigenvectors, Random Struct Algorithms 42 pp 110– (2013) · Zbl 1257.05089
[33] Wang, Random covariance matrices: Universality of local statistics of eigenvalues up to the edge, Random Matrices: Theory Appl 01 pp 1150005– (2012) · Zbl 1288.15047
[34] Wright, A bound on tail probabilities for quadratic forms in independent random variables whose distributions are not necessarily symmetric, Ann Probab 1 pp 1068– (1973) · Zbl 0271.60033
[35] Yin, On the limit of the largest eigenvalue of the large dimensional sample covariance matrix, Probab Theory Relat Fields 78 pp 509– (1988) · Zbl 0627.62022
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