  In this paper we formulate and prove a statistical version of the Candes-Tao
restricted isometry property (SRIP for short) which holds in general for any
incoherent dictionary which is a disjoint union of orthonormal bases. In
addition, we prove that, under appropriate normalization, the eigenvalues of
the associated Gram matrix fluctuate around 1 according to the Wigner
semicircle distribution. The result is then applied to various dictionaries
that arise naturally in the setting of finite harmonic analysis, giving, in
particular, a better understanding on a remark of
Applebaum-Howard-Searle-Calderbank concerning RIP for the Heisenberg dictionary
of chirp like functions.
