ON THE LIMIT BEHAVIOR OF THE JOINT DISTRIBUTION FUNCTION OF ORDER-STATISTICS

For k is-an-element-of N0 fixed we consider the joint distribution function F(n)k of the n - k smallest order statistics of n real-valued independent, identically distributed random variables with arbitrary cumulative distribution function F. The main result of the paper is a complete characterization of the limit behaviour of F(n)k(x1,....,x(n-k)) in terms of the limit behaviour of n(1-F(x(n)) if n tends to infinity, i.e., in terms of the limit superior, the limit inferior, and the limit if the latter exists. This characterization can be reformulated equivalently in terms of the limit behaviour of the cumulative distribution function of the (k + 1)-th largest order statistic. All these results do not require any further knowledge about the underlying distribution function F.