Two LIL-type results for the Brownian bridge
type: Item
instance of: scholarly article
title: Two LIL-type results for the Brownian bridge
zbMATH Open document ID: 0598.60040
DOI: 10.1016/0047-259X(86)90096-5
author: Pradip Kumar Sen
published in: Journal of Multivariate Analysis
publication date: 1986-01-01T00:00:00Z
review text: X\({}_ t\) is a Brownian bridge. Suppose H(t) is a nonnegative function defined for t close to zero such that H(t) is nondecreasing for t close to zero, and g(t) defined as \(\sqrt{t(1-t)}H(t)\) is nondecreasing for t close to zero. H is called upper class at zero if for almost all sample paths \(\omega\) there is a positive quantity \(\epsilon\) (\(\omega)\) such that for all \(t<\epsilon\), \(X_ t<g(t)\). Define \(T_ g\) as inf\(\{\) \(t: X_ t\geq g(t)\}\). The following two theorems are proved. Theorem 1. H is upper class at zero if and only if \(\int_{0+}\exp (- H^ 2(t))H(t)t^{-1}dt\) is finite. Theorem 2. If H is upper class at zero then as \(v\to 0\), \(.0907(1+o(1))I_ g(v)\leq p_ v\leq e(1+o(1))I_ g(v)\), where for \(v>0\), \(p_ v=P(T_ g\leq v)\) and \(I_ g(v)=(2\pi)^{- 1/2}\int^{v}_{0+}\exp (-H^ 2(t))H(t)t^{-1}dt\).
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 3965096
zbMATH Keywords: upper and lower class functions
zbMATH Keywords: first crossing time
zbMATH Keywords: Brownian bridge
reviewed by: Lionel Weiss
MaRDI profile type: Publication
cites work: On the Application of the Borel-Cantelli Lemma
cites work: Q3702208
cites work: On Some Asymptotic Properties Concerning Brownian Motion
description: scientific article; zbMATH DE number 3965096