Compact convex sets of the plane and probability theory∗. (29th October 2014)

• Record Type: Journal Article
• Authors: Marckert, Jean-François; Renault, David
• Is Part Of: ESAIM. Volume 18(2014)
• Page Start: 854
• Page End: 880
• Publication Date: 2014-10-29
• Subjects: Probabilities -- Periodicals; Mathematical statistics -- Periodicals; 519.2
• DOI: 10.1051/ps/2014008
• Languages: English
• ISSNs: 1292-8100

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>The Gaussâˆ'Minkowski correspondence in ℝ<sup>2</sup> states the existence of a homeomorphism between the probability measures <italic>μ</italic> on [0, 2<italic>Ï€</italic>] such that <inline-formula><alternatives><tex-math id="tex_eq1"><![CDATA[\hbox{$\int_0^{2\pi} {\rm e}^{ix}{\rm d}\mu(x)=0$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxsj7" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq1" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:msup><mml:mi /><mml:mi mathvariant="normal">∫</mml:mi></mml:msup><mml:mn mathvariant="normal">0</mml:mn><mml:mrow><mml:mn mathvariant="normal">2</mml:mn><mml:mi mathvariant="italic">Ï€</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mi mathvariant="normal">e</mml:mi><mml:mi mathvariant="italic">ix</mml:mi></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="italic">μ</mml:mi><mml:mo mathvariant="normal">(</mml:mo><mml:mi mathvariant="italic">x</mml:mi><mml:mo mathvariant="normal">)</mml:mo><mml:mo mathvariant="normal">=</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math></alternatives></inline-formula> and the compact convex sets (CCS) of the plane with perimeter 1. In this article, we bring out explicit formulas relating the border of a CCS to its probability measure. As a consequence, we show that some natural operations on CCS â€" for example, the Minkowski sum â€" have natural translations in terms of probability measure operations, and reciprocally, the convolution of measures translates into a new notion of convolution of CCS. Additionally, we give a proof that a polygonal curve associated with a sample of <italic>n</italic> random variables (satisfying <inline-formula><alternatives><tex-math id="tex_eq2"><![CDATA[\hbox{$\int_0^{2\pi} {\rm e}^{ix}{\rm d}\mu(x)=0$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxsj7" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq2" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:msup><mml:mi /><mml:mi mathvariant="normal">∫</mml:mi></mml:msup><mml:mn mathvariant="normal">0</mml:mn><mml:mrow><mml:mn mathvariant="normal">2</mml:mn><mml:mi mathvariant="italic">Ï€</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mi mathvariant="normal">e</mml:mi><mml:mi mathvariant="italic">ix</mml:mi></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="italic">μ</mml:mi><mml:mo mathvariant="normal">(</mml:mo><mml:mi mathvariant="italic">x</mml:mi><mml:mo mathvariant="normal">)</mml:mo><mml:mo mathvariant="normal">=</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math></alternatives></inline-formula>) converges to a CCS associated with <italic>μ</italic> at speed √<italic>n</italic>, a result much similar to the convergence of the empirical process in statistics. Finally, we employ this correspondence to present models of smooth random CCS and simulations.</p> </abstract>