On the convergence of random processes generated by polyhedral approximation of convex compacts

I. S. Molchanov

Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 2, pp. 438-444

Résumé

We consider a convex compact $F$ with a boundary of class $C^2$, a probability density $f$ concentrated on $F$ and continuous in some neighborhood of the boundary $\partial F$, and a random polyhedron $\Xi_n$ that coincides with a convex hull of a sample from $n$ independent points with distribution $f$. This paper studies the asymptotic behavior of a normed random process $\eta_n$ given on the unit sphere and equal to the difference of support functions of the compact $F$ and the polyhedron $\Xi _n$. The results mentioned are formulated in terms of epiconvergence, i.e., the weak convergence of epigraphs of processes as random closed sets. If $f(x)$ does not vanish at least at one point, $x\in\partial F$, then $n\Xi_n$ has a nonzero weak epi-limit as $n\to\infty$. If $f(x)=0$ on $\partial F$, but a scalar product of a gradient of $f$ and a normal to $\partial F$ is not equal to zero identically, then the right normalization would be $n^{1/2}Xi_n$. For these cases, the distributions of the limit epigraph as a closed set in the space $S^{d-1}\times\mathbf{R}$ are obtained in the paper.

Détail

Keywords: random polyhedron, convex hull, support function, random closed set, union of random sets.
Mots-clés : epiconvergence, Poisson process