The asymptotic distribution of intermediate sums. (English) Zbl 0793.60020

Csörgő, Sándor; Mason, David M.

Ann. Probab. 22, No. 1, 145-159 (1994).

Summary: Let \(X_{1,n} \leq \cdots \leq X_{n,n}\) be the order statistics of \(n\) independent random variables with a common distribution function \(F\) and let \(k_ n\) be positive numbers such that \(k_ n \to \infty\) and \(k_ n/n \to 0\) as \(n \to \infty\), and consider the sums \(I_ n(a,b)=\sum^{\lceil bk_ n \rceil}_{i= \lceil ak_ n \rceil+1} X_{n+1-i,n}\) of intermediate order statistics, where \(0<a<b\). We find necessary and sufficient conditions for the existence of constants \(A_ n>0\) and \(C_ n\) such that \(A_ n^{-1} (I_ n(a,b)-C_ n)\) converges in distribution along subsequences of the positive integers \(\{n\}\) to nondegenerate limits and completely describe the possible subsequential limiting distributions.

MSC:

60F05

Central limit and other weak theorems

Keywords:

sums of intermediate order statistics; asymptotic distribution; stochastic compactness of maxima; order statistics