Large Deviations for Proportions of Observations Which Fall in Random Sets Determined by Order Statistics

Let {X (n) :n a parts per thousand yenaEuro parts per thousand 1} be independent random variables with common distribution function F and consider , where h aaEuro parts per thousand{1,...,n}, X (1:k) a parts per thousand currency signaEuro parts per thousand a <-aEuro parts per thousand a parts per thousand currency signaEuro parts per thousand X (k:k) are the order statistics of the sample X (1),...,X (k) and D is some suitable Borel set of the real line. In this paper we prove that, if F is continuous and strictly increasing in the essential support of the distribution and if for some lambda aaEuro parts per thousand[0,1], then satisfies the large deviation principle. As a by product we derive the large deviation principle for order statistics . We also present results for the special case of Bernoulli distributed random variables with mean p aaEuro parts per thousand(0,1), and we see that the large deviation principle holds only for p a parts per thousand yenaEuro parts per thousand 1/2. We discuss further almost sure convergence of and some related quantities.