Approximation of probability density functions via location-scale finite mixtures in Lebesgue spaces

Abstract: 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.