Asymptotic properties of numbers of observations in random regions determined by central order statistics

In this paper, we extend the concept of near order statistic observation by considering observations that fall into a random region determined by a given order statistic and a Borel set. We study asymptotic properties of numbers of such observations as the sample size tends to infinity and the order statistic is a central one. We show that then proportions of these numbers converge in probability to some population probabilities. We also prove that these numbers can be centered and normalized to yield normal limit law. First, we derive results for one order statistic; next we give extensions to the multivariate case of two or more order statistics.