Fluctuations of eigenvalues and second order Poincare inequalities

• 成果类型: Article
• 署名作者: Chatterjee, Sourav
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-007-0118-6
• 发表日期: 2009
• 页码: 1-40
• 关键词: random matrices; LIMIT-THEOREMS; spectral measure; toeplitz; hankel; bounds

摘要

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 Poincare inequalities': just as ordinary Poincare inequalities give variance bounds, second order Poincare 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.