Distribution of functionals of a Ferguson–Dirichlet process over an n-dimensional ball

The c-characteristic function has been shown to have properties similar to those of the Fourier transformation. We now give a new property of the c-characteristic function of the spherically symmetric distribution. With this property, we can easily determine whether a distribution is spherically symmetric. The exact probability density function of the random mean of a spherically symmetric Ferguson–Dirichlet process with parameter measure over an n-dimensional spherical surface and that over an n-dimensional ball are given. We further give the exact probability density function of the random mean of a Ferguson–Dirichlet process with parameter measure over an n-dimensional ellipsoidal surface and that over an n-dimensional ellipsoidal solid.