## On the influence of the extremes of an i.i.d. sequence on the maximal spacings.(English)Zbl 0594.60029

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.
Reviewer: W.Dziubdziela

### MSC:

 60F05 Central limit and other weak theorems 60F15 Strong limit theorems 62G30 Order statistics; empirical distribution functions
Full Text: