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Convergence rate of expected spectral distributions of large random matrices. I: Wigner matrices. (English) Zbl 0779.60024
Let $W\sb n$ be an $n\times n$ symmetric matrix (Wigner matrix) with eigenvalues $\lambda\sb 1\le\lambda\sb 2\le\cdots\le\lambda\sb n$. Then its spectral distribution is defined by $F\sb n(x)={1\over n}\#\{i:\lambda\sb i\le x\}$, where $\#\{Q\}$ denotes the number of entries in the set $Q$. {\it 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 $O(n\sp{-1/4})$. [Part II see below].

60F15Strong limit theorems
62F15Bayesian inference
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