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Transactions of the American Mathematical Society

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Boundary crossing probabilities for stationary Gaussian processes and Brownian motion


Author: Jack Cuzick
Journal: Trans. Amer. Math. Soc. 263 (1981), 469-492
MSC: Primary 60G15; Secondary 60F10, 60J65
DOI: https://doi.org/10.1090/S0002-9947-1981-0594420-5
MathSciNet review: 594420
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Abstract: Let $ X(t)$ be a stationary Gaussian process, $ f(t)$ a continuous function, and $ T$ a finite or infinite interval. This paper develops asymptotic estimates for $ P(X(t) \geqslant f(t)$, some $ t \in T$ when this probability is small. After transformation to an Ornstein Uhlenbeck process the results are also applicable to Brownian motion. In that special case, if $ W(t)$ is Brownian motion, $ f$ is continuously differentiable, and $ T = [0,T]$ our estimate for $ P(W(t) \geqslant f(t)$, some $ t \in T)$ is

$\displaystyle \Lambda = \int_0^T {{{(2t)}^{ - 1}}(f(t)/{t^{1/2}})\phi (f(t)/{t^... ...{I_{\{ (f(t)/{t^{1/2}})'{\vert _{t = T}} < 0\} }}{\Phi ^ \ast }(f(T)/{T^{1/2}})$

provided $ \Lambda $ is small. Here $ \phi $ is the standard normal density and $ {\Phi ^ \ast }$ is its upper tail distribution. Our approach is to find an approximate first passage density and then compute crossing probabilities as a one-dimensional integral. In the case of boundaries without cusps, our results unify and extend separate results for crossings of constant levels developed by Pickands, and Qualls-Watanabe, and crossings of rapidly increasing barriers studied by Berman. Applications are also briefly explored.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0594420-5
Keywords: Boundary crossing, first passage time, Gaussian process, Brownian motion, sequential analysis, maxima
Article copyright: © Copyright 1981 American Mathematical Society

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