LIMIT THEOREMS FOR PROPORTIONS OF OBSERVATIONS FALLING INTO RANDOM REGIONS DETERMINED BY ORDER STATISTICS

Authors


  • Acknowledgements. The author thanks the anonymous referee for his careful reading of the manuscript and his detailed suggestions which led to a considerably improved version of the paper.

Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland.
 e-mail: dembinsk@mini.pw.edu.pl

Summary

In this paper, we study asymptotic behavior of proportions of sample observations that fall into random regions determined by a given Borel set and an order statistic. We show that these proportions converge almost surely to some population quantities as the sample size increases to infinity. We derive our results for independent and identically distributed observations from an arbitrary cumulative distribution function, in particular, we allow samples drawn from discontinuous laws. We also give extensions of these results to the case of randomly indexed samples with some dependence between observations.

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