Compact convex sets of the plane and probability theory

Jean-François Marckert; David Renault

ESAIM: Probability and Statistics (2014)

  • Volume: 18, page 854-880
  • ISSN: 1292-8100

Abstract

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The Gauss−Minkowski correspondence in ℝ2 states the existence of a homeomorphism between the probability measures μ on [0,2π] such that ∫ 0 2 π e i x d μ ( x ) = 0 ∫ 0 2 π e ix d μ ( x ) = 0 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 – for example, the Minkowski sum – 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 ofnrandom variables (satisfying ∫ 0 2 π e i x d μ ( x ) = 0 ∫ 0 2 π e ix d μ ( x ) = 0 ) converges to a CCS associated withμ at speed √n, 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.

How to cite

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Marckert, Jean-François, and Renault, David. "Compact convex sets of the plane and probability theory." ESAIM: Probability and Statistics 18 (2014): 854-880. <http://eudml.org/doc/274360>.

@article{Marckert2014,
abstract = {The Gauss−Minkowski correspondence in ℝ2 states the existence of a homeomorphism between the probability measures μ on [0,2π] such that $\int _0^\{2\pi \} \{\rm e\}^\{ix\}\{\rm d\}\mu (x)=0$ ∫ 0 2 π e ix d μ ( x ) = 0 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 – for example, the Minkowski sum – 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 ofnrandom variables (satisfying $\int _0^\{2\pi \} \{\rm e\}^\{ix\}\{\rm d\}\mu (x)=0$ ∫ 0 2 π e ix d μ ( x ) = 0 ) converges to a CCS associated withμ at speed √n, 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.},
author = {Marckert, Jean-François, Renault, David},
journal = {ESAIM: Probability and Statistics},
keywords = {random convex sets; symmetrisation; weak convergence; Minkowski sum; probability measure},
language = {eng},
pages = {854-880},
publisher = {EDP-Sciences},
title = {Compact convex sets of the plane and probability theory},
url = {http://eudml.org/doc/274360},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Marckert, Jean-François
AU - Renault, David
TI - Compact convex sets of the plane and probability theory
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 854
EP - 880
AB - The Gauss−Minkowski correspondence in ℝ2 states the existence of a homeomorphism between the probability measures μ on [0,2π] such that $\int _0^{2\pi } {\rm e}^{ix}{\rm d}\mu (x)=0$ ∫ 0 2 π e ix d μ ( x ) = 0 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 – for example, the Minkowski sum – 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 ofnrandom variables (satisfying $\int _0^{2\pi } {\rm e}^{ix}{\rm d}\mu (x)=0$ ∫ 0 2 π e ix d μ ( x ) = 0 ) converges to a CCS associated withμ at speed √n, 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.
LA - eng
KW - random convex sets; symmetrisation; weak convergence; Minkowski sum; probability measure
UR - http://eudml.org/doc/274360
ER -

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