Study of the Cramer-Rao bound as the numbers of observations and unknown parameters increase

For a data model consisting of deterministic signals in additive Gaussian noise, we prove that the Cramer-Rao bound (CRB) corresponding to the signal parameters decreases as the number of data samples increases provided that the number of new observations is larger than the number of additional unknowns required to parameterize these observations. We also show that the CRB theory is not applicable whenever the aforementioned condition does not hold true.