Optimal bounds for convergence of expected spectral distributions to the semi-circular law

• 成果类型: Article
• 署名作者: Goetze, F.; Tikhomirov, A.
• 署名单位: University of Bielefeld; Syktyvkar State University; Russian Academy of Sciences; Komi Science Centre of the Ural Branch of the Russian Academy of Sciences
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0629-5
• 发表日期: 2016
• 页码: 163-233
• 关键词: random matrices; inequalities; UNIVERSALITY; eigenvalues; constants

摘要

Let X = (X-jk)(j,k=1)(n) denote a Hermitian random matrix with entries X-jk, which are independent for 1 <= j <= k <= n. We consider the rate of convergence of the empirical spectral distribution function of the matrix X to the semi-circular law assuming that EX (jk) = 0, EXjk2 = 1 and that sup(n >= 1) sup(1 <= j, k <= n) E vertical bar X-jk vertical bar(4) =: mu(4) < infinity, and sup(1 <= j,k <= n) vertical bar X-jk vertical bar <= D(0)n(1/4) By means of a recursion argument it is shown that the Kolmogorov distance between the expected spectral distribution of the Wigner matrix W = 1/root nX and the semicircular law is of order O(n(-1)).