Near-Gaussian distributions for modelling discrete stellar velocity data
  with heteroskedastic uncertainties
  The velocity distributions of stellar tracers in general exhibit weak
non-Gaussianity encoding information on the orbital composition of a galaxy and
the underlying potential. The standard solution for measuring non-Gaussianity
involves constructing a series expansion (e.g. the Gauss-Hermite series) which
can produce regions of negative probability density. This is a significant
issue for the modelling of discrete data with heteroskedastic uncertainties.
Here, we introduce a method to construct positive-definite probability
distributions by the convolution of a given kernel with a Gaussian
distribution. Further convolutions by observational uncertainties are trivial.
The statistics (moments and cumulants) of the resulting distributions are
governed by the kernel distribution. Two kernels (uniform and Laplace) offer
simple drop-in replacements for a Gauss-Hermite series for negative and
positive excess kurtosis distributions with the option of skewness. We
demonstrate the power of our method by an application to real and mock
line-of-sight velocity datasets on dwarf spheroidal galaxies, where kurtosis is
indicative of orbital anisotropy and hence a route to breaking the
mass-anisotropy degeneracy for the identification of cusped versus cored dark
matter profiles. Data on the Fornax dwarf spheroidal galaxy indicate positive
excess kurtosis and hence favour a cored dark matter profile. Although designed
for discrete data, the analytic Fourier transforms of the new models also make
them appropriate for spectral fitting, which could improve the fits of high
quality data by avoiding unphysical negative wings in the line-of-sight
velocity distribution.
