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

Let A be a matrix whose columns X1, . . . ,XN are independent random vectors in Rn. Assume that the tails of the 1-dimensional marginals decay as P(| 〈Xi, a〉 | ≥ t) ≤ t−p uniformly in a ∈ Sn−1 and i ≤ N . Then for p > 4 we prove that with high probability A/ √ n has the Restricted Isometry Property (RIP) provided that Euclidean norms |Xi| are concentrated around √ n. We also show that the covariance matrix is well approximated by the empirical covariance matrix 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(−tα) with α ∈ (0, 2], extending the known case α ∈ [1, 2].