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The local geometry of finite mixtures


Authors: Elisabeth Gassiat and Ramon van Handel
Journal: Trans. Amer. Math. Soc. 366 (2014), 1047-1072
MSC (2010): Primary 41A46; Secondary 52A21, 52C17
DOI: https://doi.org/10.1090/S0002-9947-2013-06041-2
Published electronically: August 8, 2013
MathSciNet review: 3130325
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Abstract: We establish that for $ q\ge 1$, the class of convex combinations of $ q$ translates of a smooth probability density has local doubling dimension proportional to $ q$. The key difficulty in the proof is to control the local geometric structure of mixture classes. Our local geometry theorem yields a bound on the (bracketing) metric entropy of a class of normalized densities, from which a local entropy bound is deduced by a general slicing procedure.


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Additional Information

Elisabeth Gassiat
Affiliation: Laboratoire de Mathématiques, Université Paris-Sud, Bâtiment 425, 91405 Orsay Cedex, France
Email: elisabeth.gassiat@math.u-psud.fr

Ramon van Handel
Affiliation: Operations Research and Financial Engineering Department, Sherrerd Hall, Room 227, Princeton University, Princeton, New Jersey 08544
Email: rvan@princeton.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-06041-2
Keywords: Local metric entropy, bracketing numbers, finite mixtures
Received by editor(s): February 15, 2012
Received by editor(s) in revised form: August 1, 2012
Published electronically: August 8, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.