Different kinds of the invariance principle say that the uniform empirical \(\{\alpha_ n(s)\); \(0\leq s\leq 1\}\) resp. quantile process \(\{u_ n(s)\); \(0\leq s\leq 1\}\) can be approximated (in some sense) by a sequence of Brownian bridges \(\{B_ n(s)\); \(0\leq s\leq 1\}\). Let f(s) be a weight function on (0,1) with \(\lim_{s\to 0}f(s)=\lim_{s\to 1}f(s)=\infty.\) The main aim of the authors is to prove that the sequence \(\{B_ n(s)\); \(0\leq s\leq 1\}\) can be defined in such a way that \(f(s)| B_ n(s)-\alpha_ n(s)|\) resp. \(f(s)| B_ n(s)-u_ n(s)|\) should be small nearly over the whole interval (0,1).
They also investigate how an integral \(\int g(s)du_ n(s)\) can be approximated by \(\int g(s)dB_ n(s)\) where g(s) belongs to some set of real functions satisfying some regularity conditions. An analogous problem is to approximate the integral \(\int \alpha_ n(s)dQ(s)\) by the integral \(\int B_ n(s)dQ(s)\) where Q(\(\cdot)\) is the inverse of a given distribution function.