On the limit behaviour of the joint distribution function of order statistics

Fork ₀ fixed we consider the joint distribution functionF _n ^k of then-k smallest order statistics ofn real-valued independent, identically distributed random variables with arbitrary cumulative distribution functionF. The main result of the paper is a complete characterization of the limit behaviour ofF _n ^k (x ₁,,x _n-k) in terms of the limit behaviour ofn(1-F(x _n)) ifn 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 functionF.