2012/02/29 by Élisabeth Gassiat, Elisabeth Gassiat, Ramon van Handel +1 · 2 citations
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Functional Equations Stability Results #Geometry #Mathematical Dynamics and Fractals #Mathematics #math.ST #stat.TH
paper · pdf · doi:10.1090/s0002-9947-2013-06041-2
published as Trans. Amer. Math. Soc. 366, 1047-1072 (2014) · 25 pages
arxiv created 2012/08/01 · openalex publication_date 2013/08/08 · arxiv updated 2015/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We establish that for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">q≥ 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the class of convex combinations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> translates of a smooth probability density has local doubling dimension proportional to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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.