The logarithmic law of random determinant

Abstract

Consider the square random matrix $A_n=(a_{ij}{)}_{n,n}$, where $\{a_{ij}:=a_{ij}^{(n)},i,j=1,\dots ,n\}$ 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 $n\to \infty$ \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*}

Published at http://dx.doi.org/10.3150/14-BEJ615 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (3)

• The circular law for random matrices
• Random matrices: Law of the determinant
• Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles

Cited by in corpus (1)

• Exponential growth of random determinants beyond invariance