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Sequences defining a relationship between the number of parameters in a Fourier factor demand systems and the sample size such that elasticity estimates are asymptotically normal are characterized. The main technical problem in achieving this characterization is caused by the fact that the minimum eigenvalue of the expected sum of squares and cross products matrix of the generalized least squares estimator, considered as a function of the number of parameters, decreases faster than any polynomial. This problem is addressed by establishing a uniform strong law with rate for the eigenvalues of the sample sum of squares and cross products matrix. Because the minimum eingenvalue decreases faster than any polynomial, these sequences that relate parameters to sample size grow slower than any fractional power of the sample size. (MIBA)