Asymptotic properties of numbers of observations near sample quantiles

In this paper, we show that the proportions of observations falling in the left and right vicinity of the k n th order statistic converge in probability to some population quantities. We then prove that suitably normalized versions of these variables are jointly asymptotically normal under some conditions. A generalization of this result to the case of proportions of observations in the vicinity of two or more central order statistics is established next. Some concluding remarks and a potential statistical application of these results are finally made.