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.

中文翻译：

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用于模拟具有异方差不确定性的离散恒星速度数据的近高斯分布

恒星示踪剂的速度分布通常表现出关于星系轨道组成和潜在潜力的弱非高斯编码信息。测量非高斯性的标准解决方案涉及构建可以产生负概率密度区域的级数展开（例如高斯-厄米级数）。这是具有异方差不确定性的离散数据建模的一个重要问题。在这里，我们介绍了一种通过给定内核与高斯分布的卷积来构造正定概率分布的方法。由观测不确定性引起的进一步卷积是微不足道的。结果分布的统计数据（矩和累积量）由内核分布控制。两个内核（均匀和拉普拉斯）为 Gauss-Hermite 系列提供了简单的替代品，用于负和正超峰度分布，并可选择偏度。我们通过应用于矮椭球星系的真实和模拟视线速度数据集来证明我们的方法的威力，其中峰度表示轨道各向异性，因此是一种打破质量各向异性简并性以识别尖峰与核心暗物质剖面。Fornax 矮椭球星系的数据表明峰度为正，因此有利于有核的暗物质剖面。尽管是为离散数据设计的，但新模型的解析傅立叶变换也使其适用于光谱拟合，