Principes d'invariance pour la probabilité d'un dilaté de l'enveloppe convexe d'un échantillon. (Invariance principles for the probability of a dilation of the convex hull of a sample) (Q1209194)

Let \((\Omega,{\mathcal A},\text{Pr})\) be a probability space, and \(\{X_ n\}^ \infty_{n=1}\) an independent sequence of \(\mathbb{R}^ N\)-valued random variables having common distribution \(P\), which we treat as a Borel measure on \(\mathbb{R}^ N\). For each \(\omega\in\Omega\), and for \(n=1,2,\dots\), define \(C_ n^ \bullet(\omega)\) to be the closed convex hull of the set \(\{X_ 1(\omega),\dots,X_ n(\omega)\}\), and let \(C_ 0\) denote the closed convex hull of the support of \(P\). We treat \(C_ 0\) as a ``parameter'' and the random set \(C_ n^ \bullet\) as an ``estimator'' of that parameter. The author uses the random variable \(K_ n=P(\mathbb{R}^ N\backslash C_ n^ \bullet)\) to measure the ``goodness'' of the estimation. Under certain assumptions, he obtains estimates of how fast the \(K_ n\) converge to 0 in the Pr probability, and how fast \(E(K_ n)\) converges to 0. Moreover, if \(C_ n^ \bullet(\omega)\) is dilated to \((C_ n^ \bullet(\omega))^ a=\{x\in\mathbb{R}^ N\): distance\((x,C_ n^ \bullet(\omega))<a\}\), and if the number \(a\) is itself randomized in a certain way, then the convergence rates for the corresponding \(K_ n\)-type random variables \(P(\mathbb{R}^ N\backslash(C_ n^ \bullet)^ a)\) can be made independent of \(P\).