The Asymptotic Distribution of the Supremum of the Standardized Empirical Distribution Function on Subintervals

It is well known that the limit distribution of the supremum of the empirical distribution function Fₙ centered at its expectation F and standardized by division by its standard deviation is degenerate, if the supremum is taken on too large regions εₙ < F(u) < δₙ. So it is natural to look for sequences of linear transformations, so that for given sequences of sup-regions (εₙ, δₙ) the limit of the transformed sup-statistics is nondegenerate. In this paper a partial answer is given to this problem, including the case εₙ ≡ 0, δₙ ≡ 1. The results are also valid for the Studentized version of the above statistic, and the corresponding two-sided statistics are treated, too.