  An often-cited fact regarding mixing or mixture distributions is that their
density functions are able to approximate the density function of any unknown
distribution to arbitrary degrees of accuracy, provided that the mixing or
mixture distribution is sufficiently complex. This fact is often not made
concrete. We investigate and review theorems that provide approximation bounds
for mixing distributions. Connections between the approximation bounds of
mixing distributions and estimation bounds for the maximum likelihood estimator
of finite mixtures of location- scale distributions are reviewed.
