Approximation of probability distributions by convex mixtures of Gaussian measures

Abstract:

Let $\mathcal {A_{+}}=\{a=(a_{n})\in \bigcap _{p>1}\ell _{p}:a_{n}>0, \forall n\in \mathbb {N}\}$ and let $\{\phi _{j}\}_{j=1}^{\infty }$ be an enumeration of all normal distributions with mean a rational number and variance $\frac {1}{n^{2}}, n=1,2\dots$. We prove that there exists an $a\in \mathcal {A_{+}}$ such that every probability density function, continuous, with compact support in $\mathbb {R}$, can be approximated in $L^{1}$ and $L^{\infty }$ norm simultaneously by the averages $\frac {1}{\sum _{j=1}^{n}a_{j}} \sum _{j=1}^{n}a_{j}\phi _{j}$. The set of such sequences is a dense $G_{\delta }$ set in $\mathcal {A_{+}}$ and contains a dense positive cone.