  Motivated by problems in high-dimensional statistics such as mixture modeling
for classification and clustering, we consider the behavior of radial densities
as the dimension increases. We establish a form of concentration of measure,
and even a convergence in distribution, under additional assumptions. This
extends the well-known behavior of the normal distribution (its concentration
around the sphere of radius square-root of the dimension) to other radial
densities. We draw some possible consequences for statistical modeling in
high-dimensions, including a possible universality property of Gaussian
mixtures.
