Boundary crossing probabilities for stationary Gaussian processes and Brownian motion
HTML articles powered by AMS MathViewer
- by Jack Cuzick PDF
- Trans. Amer. Math. Soc. 263 (1981), 469-492 Request permission
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 \[ \Lambda = \int _0^T {{{(2t)}^{ - 1}}(f(t)/{t^{1/2}})\phi (f(t)/{t^{1/2}})} dt + {I_{\{ (f(t)/{t^{1/2}})’{|_{t = T}} < 0\} }}{\Phi ^ \ast }(f(T)/{T^{1/2}})\] 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.References
- P. Armitage, Sequential medical trials, 2nd ed., Halsted Press [John Wiley & Sons, Inc.], New York, 1975. MR 0370997
- Simeon M. Berman, Excursions of stationary Gaussian processes above high moving barriers, Ann. Probability 1 (1973), 365–387. MR 388514, DOI 10.1214/aop/1176996932
- Simeon M. Berman, Sojourns and extremes of Gaussian processes, Ann. Probability 2 (1974), 999–1026. MR 372976, DOI 10.1214/aop/1176996495
- Harald Cramér and M. R. Leadbetter, Stationary and related stochastic processes. Sample function properties and their applications, John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0217860
- Harald Cramér, M. R. Leadbetter, and R. J. Serfling, On distribution function—moment relationships in a stationary point process, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 18 (1971), 1–8. MR 290452, DOI 10.1007/BF00538483
- Xavier Fernique, Continuité des processus Gaussiens, C. R. Acad. Sci. Paris 258 (1964), 6058–6060 (French). MR 164365
- Kiyosi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der mathematischen Wissenschaften, Band 125, Springer-Verlag, Berlin-New York, 1974. Second printing, corrected. MR 0345224 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), 65-78. 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), 15-26.
- Clifford Qualls and Hisao Watanabe, Asymptotic properties of Gaussian processes, Ann. Math. Statist. 43 (1972), 580–596. MR 307318, DOI 10.1214/aoms/1177692638
- 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. Sinica 1 (1973), no. 1, 93–120. MR 321204
- James Pickands III, Upcrossing probabilities for stationary Gaussian processes, Trans. Amer. Math. Soc. 145 (1969), 51–73. MR 250367, DOI 10.1090/S0002-9947-1969-0250367-X
- V. I. Piterbarg and V. P. Prisjažnjuk, Asymptotic behavior of the probability of a large excursion for a nonstationary Gaussian process, Teor. Verojatnost. i Mat. Statist. 18 (1978), 121–134, 183 (Russian, with English summary). MR 0494458
- Eugene Seneta, Regularly varying functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin-New York, 1976. MR 0453936
- D. Siegmund, Repeated significance tests for a normal mean, Biometrika 64 (1977), no. 2, 177–189. MR 488546, DOI 10.2307/2335683
- David Slepian, The one-sided barrier problem for Gaussian noise, Bell System Tech. J. 41 (1962), 463–501. MR 133183, DOI 10.1002/j.1538-7305.1962.tb02419.x
- Michael J. Wichura, Boundary crossing probabilities associated with Motoo’s law of the iterated logarithm, Ann. Probability 1 (1973), 437–456. MR 373034, DOI 10.1214/aop/1176996938
Additional Information
- © Copyright 1981 American Mathematical Society
- 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