Probability estimates for multiparameter Brownian processes
type: Item
instance of: scholarly article
title: Probability estimates for multiparameter Brownian processes
zbMATH Open document ID: 0645.60044
author: Richard F. Bass
published in: The Annals of Probability
publication date: 1988-01-01T00:00:00Z
review text: Let F be a distribution function on \([0,1]^ d\), \(X_ i\), \(i=1,2,..\). independent random vectors with distribution function F, and \(F_ n(t)=n^{-1}\sum^{n}_{i=1}I_{[0,t]}(X_ i)\) be the empirical distribution function. Then \(\sqrt{n}(F_ n(\cdot)-F(\cdot))\) converges in law to the mean zero Gaussian process \(W_ F\) with \[ cov(W_ F(s),\quad W_ F(t))=F([0,s]\cap [0,t])-F([0,s])F([0,t]). \] If G is a function on \([0,1]^ d\), upper and lower bounds are found for \[ P\{\sup_{t\in [0,1]^ d}| W_ F(t)-G(t)| \leq \epsilon \}. \]
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 4052694
zbMATH Keywords: multiparameter Brownian sheet
zbMATH Keywords: large devitions
zbMATH Keywords: empirical distribution function
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991899
OpenAlex ID: W2035721996
DOI: 10.1214/AOP/1176991899
description: scientific article; zbMATH DE number 4052694