Embedding the finite sampling process at a rate
type: Item
instance of: scholarly article
title: Embedding the finite sampling process at a rate
zbMATH Open document ID: 0731.60026
author: Galen R. Shorack
published in: The Annals of Probability
publication date: 1991-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60G42
zbMATH DE Number: 4209207
zbMATH Keywords: random permutation
zbMATH Keywords: Brownian bridge
zbMATH Keywords: weighted in-probability approximation
zbMATH Keywords: Skorokhod embedding for martingales
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176990453
OpenAlex ID: W1983116537
DOI: 10.1214/AOP/1176990453
description: scientific article; zbMATH DE number 4209207