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

Let {aij}{aij} (1≤i,j<∞1≤i,j<∞) be i.i.d. real-valued random variables with zero mean and unit variance and let an integer sequence (Nm)m=1∞ satisfy m/Nm⟶zm/Nm⟶z for some z∈(0,1)z∈(0,1). For each m∈Nm∈N denote by AmAm the Nm×mNm×m random matrix (aij)(aij) (1≤i≤Nm1≤i≤Nm, 1≤j≤m1≤j≤m) and let sm(Am)sm(Am) be its smallest singular value. We prove that the sequence (Nm−1/2sm(Am))m=1∞ converges to 1−z almost surely. Our result does not require boundedness of any moments of aijaij'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.