On the asymptotic distribution of weighted uniform empirical and quantile processes in the middle and on the tails

Let [alpha]n and un be the uniform empirical and quantile processes. We investigate the asymptotic distribution of the suprema of [alpha]n(s)/(s(1 - s))1/2±[nu] and un(s)/(s(1 - s))1/2±[nu] with , when the supremum is taken over ranges, depending on n, in the middle of the interval [0, 1], near 0 and near 1. We show that with suitable norming factors the said asymptotic distributions can be radically different or the same, depending on the sign and value of [nu] in the weight function 1/(s(1 - s))1/2±[nu].