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

Let $A$ be a matrix whose columns $X_1,\dots ,X_N$ are independent random vectors in ${\mathbb{R}}^n$. Assume that the tails of the 1-dimensional marginals decay as $\mathbb{P}(\mathrm{\mid }⟨X_i,a⟩\mathrm{\mid }\ge t)\le t^{-p}$ uniformly in $a\in S^{n-1}$ and $i\le N$. Then for $p>4$ we prove that with high probability $A\mathrm{/}\sqrt{n}$ has the Restricted Isometry Property (RIP) provided that Euclidean norms $\mathrm{\mid }{X}_i\mathrm{\mid }$ are concentrated around $\sqrt{n}$. 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 $n\mathrm{/}N$. Moreover, we obtain sharp bounds for both problems when the decay is of the type $\exp (-t^{\alpha })$ with $\alpha \in (0,2]$,extendingtheknowncase$\alpha\in[1, 2]$.