  The symbol error rate of the minimum distance detector for an arbitrary
multi-dimensional constellation impaired by additive white Gaussian noise is
characterized as the product of a completely monotone function with a
non-negative power of the signal to noise ratio. This representation is also
shown to apply to cases when the impairing noise is compound Gaussian. Using
this general result, it is proved that the symbol error rate is completely
monotone if the rank of its constellation matrix is either one or two. Further,
a necessary and sufficient condition for the complete monotonicity of the
symbol error rate of a constellation of any dimension is also obtained.
Applications to stochastic ordering of wireless system performance are also
discussed.
