  In this paper, we aim to generalize the notion of restricted isometry
constant (RIC) in compressive sensing (CS) to restricted isometry random
variable (RIV). Associated with a deterministic encoder there are two RICs,
namely, the left and the right RIC. We show that these RICs can be generalized
to a left RIV and a right RIV for an ensemble of random encoders. We derive the
probability and the cumulative distribution functions of these RIVs for the
most widely used i.i.d. Gaussian encoders. We also derive the asymptotic
distributions of the RIVs and show that the distribution of the left RIV
converges (in distribution) to the Weibull distribution, whereas that of the
right RIV converges to the Gumbel distribution. By adopting the RIV framework,
we bring to forefront that the current practice of using eigenvalues for RIC
prediction can be improved. We show on the one hand that the eigenvalue-based
approaches tend to overestimate the RICs. On the other hand, the RIV-based
analysis yields precise estimates of the RICs. We also demonstrate that this
precise estimation aids to improve the previous RIC-based phase transition
analysis in CS.
