  We prove the convergence of the empirical spectral measure of Wishart
matrices with size-dependent entries and characterize the limiting law by its
moments. We apply our result to the cases where the entries are Bernoulli
variables with parameter c=n or truncated heavy-tailed random variables. In
both cases, when c goes to infinity or when the truncation is small, the
limiting spectrum is a perturbation of the Marchenko-Pastur distribution and we
compute its leading term.
