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Fluctuations of eigenvalues and second order Poincaré inequalities

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Abstract

Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of ‘second order Poincaré inequalities’: just as ordinary Poincaré inequalities give variance bounds, second order Poincaré inequalities give central limit theorems. The proof of the main result employs Stein’s method of normal approximation. A number of examples are worked out, some of which are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.

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Author information

Correspondence to Sourav Chatterjee.

Additional information

The author’s research was partially supported by NSF grant DMS-0707054 and a Sloan Research Fellowship.

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Chatterjee, S. Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Relat. Fields 143, 1–40 (2009). https://doi.org/10.1007/s00440-007-0118-6

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Keywords

  • Central limit theorem
  • Random matrices
  • Linear statistics of eigenvalues
  • Poincaré inequality
  • Wigner matrix
  • Wishart matrix
  • Toeplitz matrix

Mathematical Subject Classification (2000)

  • 60F05
  • 15A52