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The velocity distributions of stellar tracers in general exhibit weaknon-Gaussianity encoding information on the orbital composition of a galaxy andthe underlying potential. The standard solution for measuring non-Gaussianityinvolves constructing a series expansion (e.g. the Gauss-Hermite series) whichcan produce regions of negative probability density. This is a significantissue for the modelling of discrete data with heteroskedastic uncertainties.Here, we introduce a method to construct positive-definite probabilitydistributions by the convolution of a given kernel with a Gaussiandistribution. Further convolutions by observational uncertainties are trivial.The statistics (moments and cumulants) of the resulting distributions aregoverned by the kernel distribution. Two kernels (uniform and Laplace) offersimple drop-in replacements for a Gauss-Hermite series for negative andpositive excess kurtosis distributions with the option of skewness. Wedemonstrate the power of our method by an application to real and mockline-of-sight velocity datasets on dwarf spheroidal galaxies, where kurtosis isindicative of orbital anisotropy and hence a route to breaking themass-anisotropy degeneracy for the identification of cusped versus cored darkmatter profiles. Data on the Fornax dwarf spheroidal galaxy indicate positiveexcess kurtosis and hence favour a cored dark matter profile. Although designedfor discrete data, the analytic Fourier transforms of the new models also makethem appropriate for spectral fitting, which could improve the fits of highquality data by avoiding unphysical negative wings in the line-of-sightvelocity distribution.

Near-Gaussian distributions for modelling discrete stellar velocity data with heteroskedastic uncertainties