On the singular values of random matrices

We present an approach that allows one to bound the largest and smallest singular values of an N×n random matrix with iid rows, distributed according to a measure on Rⁿ that is supported in a relatively small ball and for which linear functionals are uniformly bounded in L_p for some p > 8, in a quantitative (non-asymptotic) fashion. Among the outcomes of this approach are optimal estimates of 1 ± c√n=N not only in the case of the above mentioned measure, but also when the measure is log-concave or when it is a product measure of iid random variables with "heavy tails".