  We revisit the problem of perturbing a large, i.i.d. random matrix by a
finite rank error. It is known that when elements of the i.i.d. matrix have
finite fourth moment, then the outlier eigenvalues of the perturbed matrix are
close to the outlier eigenvalues of the error, as long as the perturbation is
relatively small. We first prove that under a merely second moment condition,
for a large class of perturbation matrix with bounded rank and bounded operator
norm, the outlier eigenvalues of perturbed matrix still converge to that of the
perturbation. We then prove that for a matrix with i.i.d. Bernoulli $(d/n)$
entries or Bernoulli $(d_n/n)$ entries with $d_n=n^{o(1)}$, the same result
holds for perturbation matrices with a bounded number of nonzero elements.
