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August 2015 The logarithmic law of random determinant
Zhigang Bao, Guangming Pan, Wang Zhou
Bernoulli 21(3): 1600-1628 (August 2015). DOI: 10.3150/14-BEJ615

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*}

Citation

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Zhigang Bao. Guangming Pan. Wang Zhou. "The logarithmic law of random determinant." Bernoulli 21 (3) 1600 - 1628, August 2015. https://doi.org/10.3150/14-BEJ615

Information

Received: 1 May 2013; Revised: 1 January 2014; Published: August 2015
First available in Project Euclid: 27 May 2015

zbMATH: 1343.60011
MathSciNet: MR3352055
Digital Object Identifier: 10.3150/14-BEJ615

Keywords: CLT for martingale , logarithmic law , random determinant

Rights: Copyright © 2015 Bernoulli Society for Mathematical Statistics and Probability

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Vol.21 • No. 3 • August 2015
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