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Teoriya Veroyatnostei i ee Primeneniya, 1995, Volume 40, Issue 2, Pages 438–444
(Mi tvp3490)
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This article is cited in 11 scientific papers (total in 11 papers)
Short Communications
On the convergence of random processes generated by polyhedral approximation of convex compacts
I. S. Molchanov Department BS, CWI, JB Amsterdam, The Netherland
Abstract:
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.
Keywords:
random polyhedron, convex hull, support function, epiconvergence, Poisson process, random closed set, union of random sets.
Received: 16.04.1992 Revised: 18.05.1993
Citation:
I. S. Molchanov, “On the convergence of random processes generated by polyhedral approximation of convex compacts”, Teor. Veroyatnost. i Primenen., 40:2 (1995), 438–444; Theory Probab. Appl., 40:2 (1995), 383–390
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Abstract page: | 172 | Full-text PDF : | 50 | First page: | 16 |
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