Limit theorems for areas and perimeters of random inscribed and circumscribed polygons (Q2246216)

Let \(\xi _{i}\), \(i=1,2,\dots\) denote i.i.d. random variables and let \(h:\mathbb R ^{m}\rightarrow \mathbb R \) denote a symmetric Borel function. Let \(J=\left\{ (i_{1},i_{2},\dots,i_{m}):1\leq i_{1}<i_{2}<\dots<i_{m}\leq n\right\} \). Here, \(m\) is fixed and \(n\) grows unboundedly. The statistic \(U_{n}\) defined as \( U_{n}=\max_{J}h(\xi _{i_{1}},\dots,\xi _{i_{m}})\) is called a \(U\)-max statistic with kernel \(h\). In Theorem 1 the authors provide conditions under which \(U_{n}/z_{n}\) converges to a nondegenerate law. Now let \(\xi _{i}\), \(i=1,2,\dots\) denote i.i.d. random points of the circle with continuous density \(p(x)\). We set \(h_{M}(\xi _{i_{1}},\dots,\xi _{i_{m}})\) the area of the convex polygon with vertices \(\xi _{i_{1}},\dots,\xi _{i_{m}}\) and the corresponding \(U\)-max statistic is denoted by \(A_{n}^{m}\). Under suitable conditions on \(p(x)\) the authors prove a weak limit result for \(A_{n}^{m}\) as \(n\rightarrow \infty \). In Section 4 the authors prove a similar theorem for \(\tilde{A}_{n}^{m}=\min_{J}h_{M}(\xi _{i_{1}},\dots,\xi _{i_{m}})\), a \(U\)-min statistic.