Some asymptotic properties of the local time of the uniform empirical process
type: Item
instance of: scholarly article
title: Some asymptotic properties of the local time of the uniform empirical process
zbMATH Open document ID: 0960.60023
DOI: 10.2307/3318559
author: Zhan Shi
author: Marc Yor
author: Miklós Csörgo
published in: Bernoulli
publication date: 2000-04-10T00:00:00Z
review text: Let \(U_1,U_2, \dots\) denote a sequence of independent random variables, uniformly distributed on \([0,1]\), and let \(\alpha_n(t): =\sqrt n(F_n (t)-t)\), \(0\leq t\leq 1\), denote the uniform empirical process. The authors study level crossings of \(\alpha_n\), defined by \[ L_1^x(\alpha_n) =n^{-1/2} \sum_{0\leq t\leq 1}\mathbf{1}_{\{\alpha_n(t) =x\}},\quad x\in\mathbb{R}. \] The main results of the present paper are LIL-type theorems, concerning the almost sure asymptotic behavior of various functionals of \(L_1^x(\alpha_n)\), \(x\in\mathbb{R}\). These functionals include \((L_1^x (\alpha_n)\), \(\sup_x L_1^x(\alpha_n))\) and \(\int(L^x (\alpha_n))^pdx\), \(p\in\mathbb{N}\). The proofs are based on a coupling of the uniform empirical process and Brownian bridge together with tail estimates for functionals of the local time of Brownian bridge.
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 1395962
zbMATH Keywords: Brownian bridge
zbMATH Keywords: empirical process
zbMATH Keywords: local time
reviewed by: Herold Dehling
MaRDI profile type: Publication
OpenAlex ID: W2010901247
description: scientific article; zbMATH DE number 1395962