Frontiers of Mathematics in China >

2017 , Vol. 12 >Issue 4: 805 - 819

DOI: https://doi.org/10.1007/s11464-017-0629-0

RESEARCH ARTICLE

Asymptotic behavior for log-determinants of several non-Hermitian random matrices

  • Lei CHEN 1,2 ,
  • Shaochen WANG , 1,2
Expand
  • 1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
  • 2. School of Mathematics, South China University of Technology, Guangzhou 510640, China

Received date: 16 Apr 2016

Accepted date: 09 Jan 2017

Published date: 06 Jul 2017

Copyright

2017 Higher Education Press and Springer-Verlag Berlin Heidelberg
Fold

Abstract

We study the asymptotic behavior for log-determinants of two unitary but non-Hermitian random matrices: the spherical ensembles A−1B, where A and B are independent complex Ginibre ensembles and the truncation of circular unitary ensembles. The corresponding Berry-Esseen bounds and Cramér type moderate deviations are established. Our method is based on the estimates of corresponding cumulants. Numerical simulations are also presented to illustrate the theoretical results.

Key words： Log-determinants; Berry-Esseen bounds; moderate deviations; spherical ensembles; circular unitary ensembles

Cite this article

Lei CHEN , Shaochen WANG . Asymptotic behavior for log-determinants of several non-Hermitian random matrices[J]. Frontiers of Mathematics in China, 2017 , 12(4) : 805 -819 . DOI: 10.1007/s11464-017-0629-0

References

Publishing order | Descend order by publishing year | Descend order by cited within
1
AkemannG, BurdaZ. Universal microscopic correlation functions for products of independent Ginibre matrices. J Phys A, 2012, 45(46): 465201

DOI

2
BaoZ G, PanG M, ZhouW. The logarithmic law of random determinant. Bernoulli, 2015, 21(3): 1600–1628

DOI

3
CaiT, LiangT Y, ZhouH. Law of log determinant of sample covariance matrix and optimal estimation of differential entropy for high-dimensional Gaussian distributions. J Multivariate Anal, 2015, 137: 161–172

DOI

4
ChenL, GaoF Q, WangS C. Berry-esseen bounds and Cramér type large deviations for eigenvalues of random matrices. Sci China Ser A, 2015, 58(9): 1959–1980

DOI

5
DelannayR, Le CaërG. Distribution of the determinant of a random real-symmetric matrix from the Gaussian orthogonal ensemble. Phys Rev E, 2000, 62(2): 1526

DOI

6
DöringH, EichelsbacherP. Moderate deviations for the determinant of Wigner matrices. In: Limit Theorems in Probability, Statistics and Number Theory. Berlin: Springer, 2013, 253–275

DOI

7
ForresterP J, MaysA. Pfaffian point process for the Gaussian real generalized eigenvalue problem. Probab Theory Related Fields, 2012, 154(1-2): 1–47

DOI

8
GirkoV L. The central limit theorem for random determinants. Theory Probab Appl, 1980, 24(4): 729–740

DOI

9
GoodmanN R. The distribution of the determinant of a complex Wishart distributed matrix. Ann Math statist, 1963, 34(1): 178–180

DOI

10
JiangH, WangS C. Cramér type moderate deviation and Berry-Esseen bound for the Log-determinant of sample covariance matrix. Preprint

11
JiangT F, QiY C. Spectral radii of large non-hermitian random matrices. J Theoret Probab, 2015, 1–39

12
KrishnapurM. From random matrices to random analytic functions. Ann Probab, 2009, 37(1): 314–346

DOI

13
LebedevN N. Special Functions and their Applications. Englewood Cliffs: Prentice-Hall Inc, 1965

14
MezzadriF. How to generate random matrices from the classical compact groups. Notices Amer Math Soc, 2007, 54(5): 592–604

15
NguyenH, VuV. Random matrices: Law of the determinant. Ann Probab, 2014, 42(1): 146–167

DOI

16
SaulisL, StatuleviciusV A. Limit Theorems for Large Deviations. New York: Springer, 1991

DOI

17
TaoT, VuV. A central limit theorem for the determinant of a Wigner matrix. Adv Math, 2012, 231(1): 74–101

DOI

18
ZyczkowskiK. Truncations of random unitary matrices. J Phys A, 2000, 33(10): 2045

DOI

Options
Outlines

/

〈 〉