  In this work we establish the Restricted Isometry Property (RIP) of the
centered column-wise self Khatri-Rao (KR) products of $n\times N$ matrix with
iid columns drawn either uniformly from a sphere or with iid sub-Gaussian
entries. The self KR product is an $n^2\times N$-matrix which contains as
columns the vectorized (self) outer products of the columns of the original
$n\times N$-matrix. Based on a result of Adamczak et al. we show that such a
centered self KR product with independent heavy tailed columns has small RIP
constants of order $s$ with probability at least $1-C\exp(-cn)$ provided that
$s\lesssim n^2/\log^2(eN/n^2)$. Our result is applicable in various works on
covariance matching like in activity detection and MIMO gain-estimation.
