On the influence of the extremes of an i.i.d. sequence on the maximal spacings (Q1077072)

Let \(X_ 1,X_ 2,..\). be an i.i.d. sequence of random variables with a continuous density and let \(X_{1,n}\leq X_{2,n}\leq...\leq X_{n,n}\) be the order statistics of \(X_ 1,X_ 2,...,X_ n\). Define \(S_ i^{(n)}=X_{i+1,n}-X_{i,n},\quad i=1,2,...,n-1,\) \(n\geq 2\). Denote the order statistics of \(S_ 1^{(n)},S_ 2^{(n)},...,S^{(n)}_{n-1}\) by \(M^{(n)}_{n-1}\leq...\leq M_ 2^{(n)}\leq M_ 1^{(n)}.\) The random variable \(M_ k^{(n)}\) is called the k th maximal spacing. Under very general assumptions, the author shows that the weak limiting behaviour of the minimum \(X_{1,n}\) and the maximum \(X_{n,n}\) completely specifies the weak limiting behaviour of the k th maximal spacing \(M_ k^{(n)}\) as \(n\to \infty\) and he obtains the corresponding limiting distributions. As examples the cases of the normal, Cauchy, and gamma distributions are considered.