The smallest parallelepiped of \(n\) random points and peeling (Q1594521)

Let \(X_i\), \(i\geq 1\), be i.i.d. random points with uniform distributions in the \(d\)-dimensional unit cube \([0,1]^d\). The smallest parallelepiped \(A_n\) is defined as the smallest \(d\)-dimensional parallelepiped with parallel faces to the coordinate axes. The author investigates the volume \(V_n\) of \(A_n\) as \(n\to\infty\) [the exact distribution of \(V_n\) was derived by \textit{J. M. Gonzáles-Barrios}, Stat. Probab. Lett. 30, No. 2, 139-145 (1996; Zbl 0867.62045)]. Using a point process approach the author obtains the asymptotic behavior of the volumes of the \(k\)th smallest parallelepipeds \(A_n^{(k)}\) which are defined by iteration. \(A_n^{(k)}\) is defined as the smallest parallelepiped which includes all the inner points of \(A_n^{(k-1)}\); \(A_n^{(1)}= A_n\). Let \(V_n^{(k)}\) be the volume of \(A_n^{(k)}\), \(k\geq 1\). The main result of the paper is the following Theorem 1: The normalized volume \(n(1- V_n^{(k)})\) converges in distribution to \(\Gamma(2kd)\), a standard Gamma distributed random variable with parameter \(2kd\).