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Approximation of probability distributions by convex mixtures of Gaussian measures


Author: Athanassia G. Bacharoglou
Journal: Proc. Amer. Math. Soc. 138 (2010), 2619-2628
MSC (2010): Primary 62E17; Secondary 41A30
DOI: https://doi.org/10.1090/S0002-9939-10-10340-2
Published electronically: March 15, 2010
MathSciNet review: 2607892
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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.


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Additional Information

Athanassia G. Bacharoglou
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 541 24, Greece
Email: ampachar@math.auth.gr

DOI: https://doi.org/10.1090/S0002-9939-10-10340-2
Keywords: Mixture, probability density function, normal distribution, universal series, algebraic genericity.
Received by editor(s): July 15, 2009
Received by editor(s) in revised form: December 11, 2009
Published electronically: March 15, 2010
Additional Notes: This work was funded by the State Scholarships Foundation of Greece (I K Y)
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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