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 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 of n random variables (satisfying ∫ 0 2 π e ix d μ ( x ) = 0 ) converges to a CCS associated with μ 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.
Mots clés : random convex sets, symmetrisation, weak convergence, Minkowski sum
@article{PS_2014__18__854_0, author = {Marckert, Jean-Fran\c{c}ois and Renault, David}, title = {Compact convex sets of the plane and probability theory}, journal = {ESAIM: Probability and Statistics}, pages = {854--880}, publisher = {EDP-Sciences}, volume = {18}, year = {2014}, doi = {10.1051/ps/2014008}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps/2014008/} }
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 SP - 854 EP - 880 VL - 18 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2014008/ DO - 10.1051/ps/2014008 LA - en ID - PS_2014__18__854_0 ER -
%0 Journal Article %A Marckert, Jean-François %A Renault, David %T Compact convex sets of the plane and probability theory %J ESAIM: Probability and Statistics %D 2014 %P 854-880 %V 18 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2014008/ %R 10.1051/ps/2014008 %G en %F PS_2014__18__854_0
Marckert, Jean-François; Renault, David. Compact convex sets of the plane and probability theory. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 854-880. doi : 10.1051/ps/2014008. http://www.numdam.org/articles/10.1051/ps/2014008/
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