Compact convex sets of the plane and probability theory
ESAIM: Probability and Statistics, Tome 18 (2014) , pp. 854-880.

The Gauss-Minkowski correspondence in ℝ2 states the existence of a homeomorphism between the probability measures μ on [0,2π] such that 0 2π e ix dμ(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 of n random variables (satisfying 0 2π e ix dμ(x)=0 ∫ 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.

DOI : https://doi.org/10.1051/ps/2014008
Classification : 52A10,  60B05,  60D05,  60F17,  60G99
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/}
}
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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