The limit of the smallest singular value of random matrices with i.i.d. entries

Let {a_ij} (1≤i,j<∞) be i.i.d. real-valued random variables with zero mean and unit variance and let an integer sequence (N_m)_m=1^∞ satisfy m/N_m→z for some z∈(0,1). For each m∈N denote by Am the Nm×m random matrix (a_ij) (1≤i ≤ N_m, 1 ≤ j ≤ m) and let s_m(A_m) be its smallest singular value. We prove that the sequence (N_m^-1/2s_m(A_m))_m=1^∞ converges to 1-√z almost surely. Our result does not require boundedness of any moments of aij's higher than the 2-nd and resolves a long standing question regarding the weakest moment assumptions on the distribution of the entries sufficient for the convergence to hold.