Quadratic functionals and small ball probabilities for the \(m\)-fold integrated Brownian motion
type: Item
instance of: scholarly article
title: Quadratic functionals and small ball probabilities for the \(m\)-fold integrated Brownian motion
zbMATH Open document ID: 1030.60026
author: Xia Chen
author: Wenbo V. Li
published in: The Annals of Probability
publication date: 2004-02-08T00:00:00Z
review text: Let \(W(t)\), \(t\geq 0\), be a standard Brownian motion starting at \(0\). Denote by \(X_0(t)= W(t)\), \(X_m(t)= \int^t_0 X_{m-1}(s) ds\), \(t\geq 0\), \(m\geq 1\), the \(m\)-fold integrated Brownian motion for positive integer \(m\). This paper is concerned with the quadratic functionals for the real process \(X_m(t)\) as well as the asymptotic behavior of the small ball probability. A method for estimating the small probability such as \(P(\sup_{0\leq i\leq 1}|X_m(t)|\leq\varepsilon)\), as \(\varepsilon\to 0\), is developed via the appropriate quadratic functionals.
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 1933040
zbMATH Keywords: Brownian motion
zbMATH Keywords: small ball probabilities
reviewed by: Ion C. Vladimirescu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1048516545
OpenAlex ID: W1963783631
cites work: The Integral of a Symmetric Unimodal Function over a Symmetric Convex Set and Some Probability Inequalities
cites work: Q4726487
cites work: Q3036394
cites work: Exact formulas of certain functional integrals on wiener spaces
cites work: Q3906660
cites work: Q5690466
cites work: Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm
cites work: Fubini's theorem for double Wiener integrals and the variance of the Brownian path
cites work: On the principal eigenvalue of second-order elliptic differential operators
cites work: On the lower tail of Gaussian seminorms
cites work: Q3721531
cites work: Q3409967
cites work: Q4032143
cites work: Q3721264
cites work: Integrated Brownian motion, conditioned to be positive
cites work: Chung’s law for integrated Brownian motion
cites work: Calculation of Characteristic Functions of Some Functionals of a Wiener Process and a Brownian Bridge
cites work: Zufällige Bewegungen. (Zur Theorie der Brownschen Bewegung.)
cites work: Metric entropy and the small ball problem for Gaussian measures
cites work: The Gaussian measure of shifted balls
cites work: Lim inf results for Gaussian samples and Chung's functional LIL
cites work: Local asymptotic classes for the successive primitives of Brownian motion
cites work: Regular points for the successive primitives of Brownian motion
cites work: Some martingales related to the integral of Brownian motion. Applications to the passage times and transience
cites work: Q4416879
cites work: Small ball probabilities for Gaussian Markov processes under the \(L_p\)-norm.
cites work: Existence of small ball constants for fractional Brownian motions
cites work: Approximation, metric entropy and small ball estimates for Gaussian measures
cites work: Gaussian processes: Inequalities, small ball probabilities and applications
cites work: Path properties of the primitives of a Brownian motion
cites work: A winding problem for a resonator driven by a white noise
cites work: Radon-Nikodym Derivatives of Gaussian Measures
cites work: Q4194315
cites work: The asymptotic expansion of integral functions defined by Taylor series
cites work: Q4693741
DOI: 10.1214/AOP/1048516545
description: scientific article; zbMATH DE number 1933040