  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 $\mathbb{P}(|\langle X_i, a\rangle|\geq t)\leq t^{-p}$ uniformly in
$a\in S^{n-1}$ and $i\leq N$. Then for $p>4$ 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^{\alpha}})$ with $\alpha
\in (0,2]$, extending the known case $\alpha\in[1, 2]$.
