Limit theorems for the number of near order statistic observations

In the present report, limit theorems for the number of near order statistics observations are discussed. Asymptotic results for the number of maxima are viewed in the discrete case. Limit theorems for their continues analogues, the numbers of near order statistic observations, are also presented. It will be shown that the last variables have close links with spacings. In our report, we also discuss the asymptotic behaviour of the ratio of order statistics. The above results are applied for analyzing runs based on spacings.