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December, 1976 Distribution Estimates for Functionals of the Two-Parameter Wiener Process
Victor Goodman
Ann. Probab. 4(6): 977-982 (December, 1976). DOI: 10.1214/aop/1176995940

Abstract

Bounds on absorption probabilities for Banach space-valued Brownian motion are obtained as expectations of estimates for the conditional probability given the endpoint of the path. The results are applied to the problem of computing the tail distributions of the supremum, $S$, of the two-parameter Wiener process and the supremum, $S'$, of its tied-down version. It is shown that for $\lambda \geqq 0$, $$P\{S' \geqq \lambda\} \geqq (2\lambda^2 + 1)\exp\lbrack -2\lambda^2 \rbrack$$ and $$P\{S \geqq \lambda\} \geqq 4 \int^\infty_\lambda sN(-s)ds$$ where $N(s)$ denotes the standard normal distribution. A corollary is that $P(S \geqq \lambda) \approx 4N(-\lambda)$ as $\lambda \rightarrow + \infty$.

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Victor Goodman. "Distribution Estimates for Functionals of the Two-Parameter Wiener Process." Ann. Probab. 4 (6) 977 - 982, December, 1976. https://doi.org/10.1214/aop/1176995940

Information

Published: December, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0344.60048
MathSciNet: MR423556
Digital Object Identifier: 10.1214/aop/1176995940

Subjects:
Primary: 60J65
Secondary: 60G15 , 60J70 , 62H10

Keywords: $N$-parameter Wiener process , Abstract Wiener space , Brownian motion , Brownian sheets , Cameron-Yeh process

Rights: Copyright © 1976 Institute of Mathematical Statistics

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Vol.4 • No. 6 • December, 1976
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