Boundary crossing probabilities for stationary Gaussian processes and Brownian motion
Author:
Jack Cuzick
Journal:
Trans. Amer. Math. Soc. 263 (1981), 469492
MSC:
Primary 60G15; Secondary 60F10, 60J65
MathSciNet review:
594420
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Abstract: Let be a stationary Gaussian process, a continuous function, and a finite or infinite interval. This paper develops asymptotic estimates for , some 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 is Brownian motion, is continuously differentiable, and our estimate for , some is provided is small. Here is the standard normal density and is its upper tail distribution. Our approach is to find an approximate first passage density and then compute crossing probabilities as a onedimensional integral. In the case of boundaries without cusps, our results unify and extend separate results for crossings of constant levels developed by Pickands, and QuallsWatanabe, and crossings of rapidly increasing barriers studied by Berman. Applications are also briefly explored.
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 Ju. V. Kozacěnko and V. I. Rudomanov, On the probability that a stationary random process exceeds a given function, Theory Prob. Math. Statist. 12 (1976), 6578.
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 C. K. McPherson and P. Armitage, Repeated significance tests on accumulating data when the null hypothesis is not true, J. Roy. Statist. Soc. Ser. A 134 (1971), 1526.
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 C. Quails and H. Watanabe, Asymptotic properties of Gaussian processes, Ann. Inst. Math. Statist. 43 (1972), 580596. MR 0307318 (46:6438)
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 H. Robbins and D. Siegmund, Statistical tests of power one and the integral representation of solutions of certain partial differential equations, Bull. Inst. Math. Acad. Síca 1 (1973), 93120. MR 0321204 (47:9737)
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 E. Seneta, Regularly varying functions, Lecture Notes in Math., vol. 508, Springer, Berlin, 1976. MR 0453936 (56:12189)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198105944205
PII:
S 00029947(1981)05944205
Keywords:
Boundary crossing,
first passage time,
Gaussian process,
Brownian motion,
sequential analysis,
maxima
Article copyright:
© Copyright 1981
American Mathematical Society
