Let δ > 1 and β > 0 be some real numbers. We prove that there are positive u, v, N 0 depending only on β and δ with the following property: for any N,n such that N ≥ max(N 0 , δ n ), any N × n random matrix A = (a ij ) with i.i.d. entries satisfying (Formula presented.) and any non-random N × n matrix B, the smallest singular value s n of A + B satisfies (Formula presented.). The result holds without any moment assumptions on the distribution of the entries of A.
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