Asymptotic behavior for log-determinants of several non-Hermitian rand om matrices

doi:10.1007/s11464-017-0629-0

Front. Math. China

2017, Vol. 12

Issue (4) : 805-819 https://doi.org/10.1007/s11464-017-0629-0

RESEARCH ARTICLE

Asymptotic behavior for log-determinants of several non-Hermitian rand om matrices

Lei CHEN^1,², Shaochen WANG^1,²(

)

¹. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
². School of Mathematics, South China University of Technology, Guangzhou 510640, China

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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.

Keywords Log-determinants Berry-Esseen bounds moderate deviations spherical ensembles circular unitary ensembles

Corresponding Author(s): Shaochen WANG

Issue Date: 06 July 2017

Cite this article:

Lei CHEN,Shaochen WANG. Asymptotic behavior for log-determinants of several non-Hermitian rand om matrices[J]. Front. Math. China, 2017, 12(4): 805-819.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-017-0629-0
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I4/805

1	AkemannG, BurdaZ. Universal microscopic correlation functions for products of independent Ginibre matrices. J Phys A, 2012, 45(46): 465201 https://doi.org/10.1088/1751-8113/45/46/465201
2	BaoZ G, PanG M, ZhouW. The logarithmic law of random determinant. Bernoulli, 2015, 21(3): 1600–1628 https://doi.org/10.3150/14-BEJ615
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 https://doi.org/10.1016/j.jmva.2015.02.003
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 https://doi.org/10.1007/s11425-014-4948-2
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 https://doi.org/10.1103/PhysRevE.62.1526
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 https://doi.org/10.1007/978-3-642-36068-8_12
7	ForresterP J, MaysA. Pfaffian point process for the Gaussian real generalized eigenvalue problem. Probab Theory Related Fields, 2012, 154(1-2): 1–47 https://doi.org/10.1007/s00440-011-0361-8
8	GirkoV L. The central limit theorem for random determinants. Theory Probab Appl, 1980, 24(4): 729–740 https://doi.org/10.1137/1124086
9	GoodmanN R. The distribution of the determinant of a complex Wishart distributed matrix. Ann Math statist, 1963, 34(1): 178–180 https://doi.org/10.1214/aoms/1177704251
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 https://doi.org/10.1214/08-AOP404
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 https://doi.org/10.1214/12-AOP791
16	SaulisL, StatuleviciusV A. Limit Theorems for Large Deviations. New York: Springer, 1991 https://doi.org/10.1007/978-94-011-3530-6
17	TaoT, VuV. A central limit theorem for the determinant of a Wigner matrix. Adv Math, 2012, 231(1): 74–101 https://doi.org/10.1016/j.aim.2012.05.006
18	ZyczkowskiK. Truncations of random unitary matrices. J Phys A, 2000, 33(10): 2045 https://doi.org/10.1088/0305-4470/33/10/307

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