The covariance matrix of the Fourier coefficients ofN- sampled stationary random signals is studied. Three theorems are established. 1) If the covariance sequence is summable, the magnitude of every off-diagonal covariance element converges to zero asN \rightarrow \infty. 2) If the covariance sequence is only square summable, the magnitude of the covariance elements sufficiently far from the diagonal converges to zero asN \rightarrow \infty. 3) If the covariance sequence is square summable, the weak norm of the matrix containing only the off-diagonal elements converges to zero asN \rightarrow \infty. The rates of convergence are also determined when the covariance sequence satisfies additional conditions.