Convergence rate of expected spectral distributions of large random matrices. I: Wigner matrices. (English) Zbl 0779.60024

Bai, Z. D.

Ann. Probab. 21, No. 2, 625-648 (1993).

Let \(W_ n\) be an \(n\times n\) symmetric matrix (Wigner matrix) with eigenvalues \(\lambda_ 1\leq\lambda_ 2\leq\cdots\leq\lambda_ n\). Then its spectral distribution is defined by \(F_ n(x)={1\over n}\#\{i:\lambda_ i\leq x\}\), where \(\#\{Q\}\) denotes the number of entries in the set \(Q\). E. P. Wigner [Ann. Math., II. Ser. 62, 548- 564 (1955; Zbl 0067.084) and ibid. 67, 325-327 (1958; Zbl 0085.132)] proved that the expected spectral distribution of Wigner matrix tends to the so-called semicircular law.
The paper develops a new methodology to establish convergence rates of empirical spectral distributions of high-dimensional random matrices. The rate of convergence for Wigner matrices established in the paper is \(O(n^{-1/4})\). [Part II see below].

Reviewer: V.Sakalauskas (Kaunas)

MSC:

60F15	Strong limit theorems
62F15	Bayesian inference

Keywords:

distribution of Wigner matrix; semicircular law; convergence rates; empirical spectral distributions

Citations:

Zbl 0067.084; Zbl 0085.132; Zbl 0779.60025