Approximation by finite mixtures of continuous density functions that vanish at infinity
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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 C_0 can be used to conduct approximation of various classes of approximands in a number of different modes. That is, we prove approximands in C_0 can be uniformly approximated, approximands in C_b can be uniformly approximated on compact sets, and approximands in L_p can be approximated with respect to the L_p, for p∈[1,∞). Furthermore, we also prove that measurable functions can be approximated, almost everywhere.
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