  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.
