Distributions of tail empirical processes on Banach function spaces (Q1324889)

The proof of \(L_ p\) norms of weighted uniform empirical processes relies heavily on a delicate construction of the Brownian bridge approximating the empirical process in a weighted metric, obtained by \textit{M. Csörgő, S. Csörgő, L. Horváth} and \textit{D. M. Mason} [Ann. Probab. 14, 31-85 (1986; Zbl 0589.60029)]. The question we are going to answer is whether the proof can also be given using results and techniques from the field of probability in Banach spaces. In a slightly simpler case, an attempt to answer this question was undertaken in [author, Stochastic Processes Appl. 46, No. 1, 1-27 (1993; Zbl 0780.60011)]. There a weak convergence of distributions on Orlicz spaces induced by empirical and quantile processes was characterized. In this paper, as a path space, we consider a Banach function space of measurable functions defined on a finite measure space \(({\mathbb{R}}_ +,m)\). Moreover, in addition, we ask: assuming the fact that the weak convergence of distributions on a path space has been proved, can we then construct the limiting stochastic process?