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

Abstract: Let $A$ be a matrix whose columns $X_1,\dots, X_N$ are independent random vectors in $\mathbb{R}<sup>n$. Assume that the tails of the 1-dimensional marginals decay as $$\mathbb{P}(|\langle X_i, a\rangle|\geq t)\leq t<sup>{-p}$$ uniformly in $a\in S<sup>{n-1}$ and $i\leq N$. Then for $p&gt;4$ we prove that with high probability $A/{\sqrt{n}}$ has the Restricted Isometry Property (RIP) provided that Euclidean norms $|X_i|$ 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 $n/N$. Moreover, we obtain sharp bounds for both problems when the decay is of the type $\exp({-t<sup>{\alpha}})$ with $\alpha \in (0,2]$, extending the known case $\alpha\in[1, 2]$.