Embedding the finite sampling process at a rate
For every \(n\geq 1\), let \(c_{n1},...,c_{nn}\) be real constants which are normalized such that \(\sum^{n}_{i=1}c_{ni}/n=0\) and \(\sum^{n}_{i=1}c^ 2_{ni}/n=1\), and let \((D_{n1},...,D_{nn})\) denote a random permutation of \((1,...,n)\) in which each of the n! permutations are equally likely. Define the finite sampling process \(R_ n=\{R_ n(t):\;0\leq t\leq 1\}\) by setting \[ R_ n(t)=n^{- 1/2}\sum^{[nt]}_{i=1}c_{nD_{ni}}, \] and let W be a Brownian bridge. It is shown that if \(\sup_{n\geq 1}\sum^{n}_{i=1}c^ 4_{ni}/n<\infty\), then the random permutations \(D_{n1},...,D_{nn}\) can be defined in such a way that for each \(0\leq \nu <1/4\) \[ \sup_{1/(n+1)\leq t\leq n/(n+1)}| (R_ n(t)-W(t))/(t(1-t))^{- t}| =O_ p(n^{-\nu}). \] Moreover, the same type of weighted in- probability approximation also holds with \(R_ n\) replaced by a weighted uniform empirical process with weights \(c_{ni}\). The proof exploits a martingale structure of the process \(R_ n\) combined with the Skorokhod embedding for martingales.
scientific article; zbMATH DE number 4209207