The Annals of Probability

The Two-Parameter Brownian Bridge: Kolmogorov Inequalities and Upper and Lower Bounds for the Distribution of the Maximum

Enrique M. Cabana and Mario Wschebor

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Abstract

The aim of this paper is to give upper and lower bounds for the probability density at $(u - z)$ of the position at time $(x, y) (x, y, z, u \in R^+)$ of a standard Wiener process with two-dimensional parameter $(x, y)$ with the requirement that it did not reach the barrier $u$ in the "past" $\{(x', y'): 0 \leq x' \leq x, 0 \leq y' \leq y\}$. The fundamental tools are Kolmogorov forward inequalities for the density and certain bounds for the behaviour of $p$ near the border.

Article information

Source
Ann. Probab., Volume 10, Number 2 (1982), 289-302.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993858

Digital Object Identifier
doi:10.1214/aop/1176993858

Mathematical Reviews number (MathSciNet)
MR647505

Zentralblatt MATH identifier
0532.60072

JSTOR
links.jstor.org

Subjects
Primary: 60G99: None of the above, but in this section
Secondary: 62G10: Hypothesis testing

Keywords
Two-parameter Brownian Bridge Kolmogorov inequalities distribution of the maximum heat equation

Citation

Cabana, Enrique M.; Wschebor, Mario. The Two-Parameter Brownian Bridge: Kolmogorov Inequalities and Upper and Lower Bounds for the Distribution of the Maximum. Ann. Probab. 10 (1982), no. 2, 289--302. doi:10.1214/aop/1176993858. https://projecteuclid.org/euclid.aop/1176993858


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