The class of location-scale finite mixtures is of enduring interest both from applied and theoretical perspectives of probability and statistics. We prove the following results: to an arbitrary degree of accuracy, (a) location-scale mixtures of a continuous probability density function (PDF) can approximate any continuous PDF, uniformly, on a compact set; and (b) for any finite $p\ge1$, location-scale mixtures of an essentially bounded PDF can approximate any PDF in $\mathcal{L}_{p}$, in the $\mathcal{L}_{p}$ norm.

翻译：从应用和理论角度的概率和统计角度来看,定位尺度有限混合物的类别具有持久的兴趣。我们证明以下结果:(a) 任意程度的精确度,(a) 连续概率密度函数(PDF)的定位尺度混合物,可以统一地在一套紧凑材料上接近任何连续的 PDF ;以及(b) 对于任何限定的 $p\ge1$,基本上受约束的PDF 的定位尺度混合物,可以以$\mathcal{L ⁇ _p}$的规范,以$\mathcal{L ⁇ p}$接近任何 PDF 。