Random matrices: Law of the determinant

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Let Aₙ be an n by n random matrix whose entries are independent real random variables with mean zero, variance one and with subexponential tail. We show that the logarithm of | Aₙ| satisfies a central limit theorem. More precisely, eqnarray*x∈ R|P((| Aₙ|)-(1/2) (n-1)!√(1/2) n≤ x)-P(N(0,1)≤ x)|≤-1/3+o(1)n.eqnarray*