An upper bound on the smallest singular value of a square random matrix

Let A=(aij) be a square n×n matrix with i.i.d. zero mean and unit variance entries. It was shown by Rudelson and Vershynin in 2008 that the upper bound for the smallest singular value sn(A) is of order n−12 with probability close to one under the additional assumption that the entries of A satisfy Ea114<∞. We remove the assumption on the fourth moment and show the upper bound assuming only Ea112=1.