In this paper, it is shown that the symbol error rate of an arbitrary multi-dimensional constellation impaired by independent and identically distributed additive white Gaussian noise under maximum likelihood detection is completely monotonic in the average SNR, if the rank of the constellation matrix is either one or two. Further, under minimum distance detection, this result generalizes to the case when the noise follows a dependent and identically distributed compound Gaussian distribution. Classes of constellations with dimensionality greater than two, which have completely monotone and convex error rates are also identified herein. The applications of the complete monotonicity of the symbol error rate in obtaining generic comparisons of average error rates over pairs of fading channels are also discussed.