A Refinement of the Central Limit Theorem for Random Determinants

The paper proves the central limit theorem (the logarithmic law) for random determinants under weaker conditions than the author used earlier: if for any n the random elements ξᵢⱼ⁽ⁿ⁾, i,j=1 n, of the matrix Ξ=(ξᵢⱼ/n) are independent, ξᵢⱼ⁽ⁿ⁾=a, ξᵢⱼ⁽ⁿ⁾=1, and for some > 0 ₙmaxi,j=1 n||⁴⁺ < , then $$ %{ \{{logΞ^2-log (n-1)!-log (1+na^2) √2log n} < x\} %& ={1√2π}∫₋^xexp(-{u^2 2})\,du.