Approximation by finite mixtures of continuous density functions that vanish at infinity
TrungTin Nguyen, Hien D. Nguyen, Faïcel Chamroukhi, Geoffrey J. McLachlan
Abstract
Given sufficiently many components, it is often cited that finite mixture models can approximate any other probability density function (pdf) to an arbitrary degree of accuracy. Unfortunately, the nature of this approximation result is often left unclear. We prove that finite mixture models constructed from pdfs in 0 can be used to conduct approximation of various classes of approximands in a number of different modes. That is, we prove approximands in 0 can be uniformly approximated, approximands in b can be uniformly approximated on compact sets, and approximands in ℒp can be approximated with respect to the ℒp , for p∈[1,∞) . Furthermore, we also prove that measurable functions can be approximated, almost everywhere.