We prove that iid random vectors that satisfy a rather weak moment assumption can
be used as measurement vectors in Compressed Sensing, and the number of measurements
required for exact reconstruction is the same as the best possible estimate -- exhibited
by a random gaussian matrix. We also prove that this moment condition is necessary,
up to a \(\log \log \) factor. Applications to the Compatibility Condition and the Restricted
Eigenvalue Condition in the noisy setup and to properties of neighbourly random polytopes
are also discussed.