A classical problem in digital communications is to evaluate the symbol error probability (SEP) and bit error probability (BEP) of a multidimensional constellation over an additive white Gaussian noise channel. Exact closed-form asymptotic expressions for SEP and BEP for arbitrary constellations and input distributions are presented. The well-known union bound is proven to be asymptotically tight under general conditions. Necessary and sufficient conditions for asymptotic equivalence between the optimal maximum a posteriori (MAP) detector and the practically relevant maximum likelihood (ML) detector are also presented.