The Annals of Probability

Necessary and Sufficient Conditions for Almost Sure Convergence of the Largest Eigenvalue of a Wigner Matrix

Z. D. Bai and Y. Q. Yin

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Abstract

Let $W = (X_{ij}; 1 \leq i, j < \infty)$ be an infinite matrix. Suppose $W$ is symmetric, entries on the diagonal are $\operatorname{iid}$, entries off the diagonal are $\operatorname{iid}$ and they are independent. Then it is proved that the necessary and sufficient conditions for $\lambda_{\max}((1/\sqrt{n})W_n) \rightarrow a \mathrm{a.s.}$ are (1) $E(X^+_{11})^2 < \infty$; (2) $EX^4_{12} < \infty$; (3) $EX_{12} \leq 0$; (4) $a = 2\sigma, \sigma^2 = EX^2_{12}$. Here $W_n = (X_{ij}; 1 \leq i, j \leq n)$.

Article information

Source
Ann. Probab., Volume 16, Number 4 (1988), 1729-1741.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991594

Digital Object Identifier
doi:10.1214/aop/1176991594

Mathematical Reviews number (MathSciNet)
MR958213

Zentralblatt MATH identifier
0677.60038

JSTOR
links.jstor.org

Subjects
Primary: 60F99: None of the above, but in this section
Secondary: 62E20: Asymptotic distribution theory

Keywords
Random matrix Wigner matrix largest eigenvalue semicircle law

Citation

Bai, Z. D.; Yin, Y. Q. Necessary and Sufficient Conditions for Almost Sure Convergence of the Largest Eigenvalue of a Wigner Matrix. Ann. Probab. 16 (1988), no. 4, 1729--1741. doi:10.1214/aop/1176991594. https://projecteuclid.org/euclid.aop/1176991594


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