On the interval of fluctuation of the singular values of random matrices

Let A be a matrix whose columns X-1, . . . , X-N are independent random vectors in R n. Assume that the tails of the 1-dimensional marginals decay as P(vertical bar < X-i ,a >vertical bar >= t) <= t(-p) uniformly in a is an element of Sn-1 and i <= N. Then for p > 4 we prove that with high probability A/root n has the Restricted Isometry Property (RIP) provided that the Euclidean norms vertical bar X-i vertical bar are concentrated around root n. We also show that the covariance matrix is well approximated by empirical covariance matrices and establish corresponding quantitative estimates on the rate of convergence in terms of the ratio n/N. Moreover, we obtain sharp bounds for both problems when the decay is of the type exp (-t(alpha)) with alpha is an element of[0,2], extending the known case alpha is an element of[1,2].