A note on the number of observations near an order statistic

Let X1,…,Xn be a sequence of independent identically distributed random variables with some continuous distribution function F. Let X1,n⩽⋯⩽Xn,n be the order statistics generated by the sample. Denote the number of observations registered in the random interval (Xn-k,n-a,Xn-k,n] by K-(n,k,a) if a>0 and by K+(n,k,a) if a<0. Some limit results for K- and K+ are established in this paper. The asymptotic independence of neighboring spacings is proved for some class of continuous distributions F.