Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles

• Published in: Probability theory and related fields Vol. 144; no. 1-2; pp. 221 - 246
• Main Author: Jiang, Tiefeng
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-05-2009; Springer; Springer Nature B.V
• Subjects: Approximation; Central limit theorem; Complex variables; Economics; Eigenvalues; Exact sciences and technology; Finance; General topics; Independent variables; Insurance; Limit theorems; Management; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Matrix; Normal distribution; Norms; Operations Research/Decision Theory; Probability; Probability and statistics; Probability Theory and Stochastic Processes; Quantitative Finance; Random variables; Sciences and techniques of general use; Statistics for Business; Studies; Theoretical; Theoretical mathematics; 15A52; Random matrix; Haar measure; 15A33; Eigenvalue; Limiting distribution; Empirical distribution; Largest eigenvalue; 60F05; 60F15; Approximation; Probability theory; Central limit theorem; Limit distribution; Law of large numbers; Random variable
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-008-0146-x

We develop a tool to approximate the entries of a large dimensional complex Jacobi ensemble with independent complex Gaussian random variables. Based on this and the author’s earlier work in this direction, we obtain the Tracy–Widom law of the largest singular values of the Jacobi emsemble. Moreover, the circular law, the Marchenko–Pastur law, the central limit theorem, and the laws of large numbers for the spectral norms are also obtained.