## Bernoulli

• Bernoulli
• Volume 21, Number 3 (2015), 1600-1628.

### The logarithmic law of random determinant

#### Abstract

Consider the square random matrix $A_{n}=(a_{ij})_{n,n}$, where $\{a_{ij}:=a_{ij}^{(n)},i,j=1,\ldots,n\}$ is a collection of independent real random variables with means zero and variances one. Under the additional moment condition $\sup_{n}\max_{1\leq i,j\leq n}\mathbb{E}a_{ij}^{4}<\infty,$ we prove Girko’s logarithmic law of $\det A_{n}$ in the sense that as $n\rightarrow\infty$ \begin{eqnarray*}\frac{\log|\det A_{n}|-(1/2)\log(n-1)!}{\sqrt{(1/2)\log n}}\stackrel{d}{\longrightarrow}N(0,1).\end{eqnarray*}

#### Article information

Source
Bernoulli, Volume 21, Number 3 (2015), 1600-1628.

Dates
Revised: January 2014
First available in Project Euclid: 27 May 2015

https://projecteuclid.org/euclid.bj/1432732031

Digital Object Identifier
doi:10.3150/14-BEJ615

Mathematical Reviews number (MathSciNet)
MR3352055

Zentralblatt MATH identifier
1343.60011

#### Citation

Bao, Zhigang; Pan, Guangming; Zhou, Wang. The logarithmic law of random determinant. Bernoulli 21 (2015), no. 3, 1600--1628. doi:10.3150/14-BEJ615. https://projecteuclid.org/euclid.bj/1432732031

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