CONCENTRATION OF THE SPECTRAL MEASURE FOR LARGE MATRICES

By Alice Guionnet, Ofer Zeitouni. 2000

237 citations recorded by the source corpus.

We derive concentration inequalities for functions of the empirical measure of eigenvalues for large, random, self adjoint matrices, with not necessarily Gaussian entries. The results presented apply in particular to non-Gaussian Wigner and Wishart matrices. We also provide concentration bounds for non commutative functionals of random matrices.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/bcbb0851d207720a9dd048b4be9f683ed7743f67

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In Mathematical Physics

1. Jun Yin
2. L. Erdős
3. Horng-Tzer Yau
4. Antti Knowles
5. Terence Tao
6. Van H. Vu
7. Debashis Paul
8. Ji Oon Lee

In Statistics and Probability

1. Jun Yin
2. L. Erdős
3. Horng-Tzer Yau
4. Antti Knowles
5. Terence Tao
6. Van H. Vu
7. Debashis Paul
8. Alexander Aue

Citations in the corpus

• Free transport-entropy inequalities for non-convex potentials and application to concentration for random matrices (2012, background): “We want to emphasise that while there are many concentration results for families of Wigner or Wishart type matrices (the interested reader can consult for example [GZ00] or chapter 4.”
• ASYMPTOTICS OF SAMPLE EIGENSTRUCTURE FOR A LARGE DIMENSIONAL SPIKED COVARIANCE MODEL (2007, methodology): “A concentration inequality for Lipschitz functionals We restate Corollary 1.8(b) of Guionnet and Zeitouni (2000) in our context.”
• A Dynamical Approach to Random Matrix Theory (2017, methodology): “Various improvements were made to shrink the spectral windows; in particular, results down to scale N−1/2 were obtained by combining the results of [11] and [79].”
• Fluctuations of eigenvalues and second order Poincaré inequalities (2007, background): “Even exponentially decaying tail bounds are available [27].”
• Large deviations upper bounds and central limit theorems for non-commutative functionals of Gaussian large random matrices (2002, background): “As in [14], we observe that W N is related to band matrices as follows.”
• Random covariance matrices: Universality of local statistics of eigenvalues (2009, background): “We isolate one particular consequence of Theorem 20 (also established in [17]):”
• Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order (2013, background): “2 to additive functionals and U -statistics of random vectors, in particular to linear eigenvalue statistics of random matrices, obtaining bounds which complement known estimates by Guionnet and Zeitouni [31].”
• On the distribution of SINR for the MMSE MIMO receiver and performance analysis (2006, background): “= p(2) m2 2 kE tr( (2)) p 1 + p(2) m2 2 k(p 1)Var tr( ) p 1 ! c(2) kM 0 J ( 1) because E( tr( ) p 1 )!M 0 J ( 1) by the bounded convergence theorem, and (p 1)Var( tr( ) p 1 )!0, which can be proved using the results from the concentration of spectral measures for random matrices [32].”
• Large deviations for the largest eigenvalue of Rademacher matrices (2018, methodology): “For this, we will use two concentration results respectively from [16] and [3].”
• Random matrix theory in statistics: A review (2014, background): “: ð25Þ A survey of such concentration inequalities for a wide class of random matrices can be found in Anderson et al. (2009), Guionnet (2009) and Guionnet and Zeitouni (2000).”
• Around the circular law (2011, unknown): “Guionnet and Zeitouni have also obtained the convergence of the support in a more recent work [73].”
• Universality of local eigenvalue statistics for some sample covariance matrices (2005, methodology): “This will be obtained using conce ntration results for the spectral measure of large random matrices due to [14].”