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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

**[1]**P. Armitage,*Sequential medical trials*, 2nd ed., Blackwell, Oxford, 1975. MR**0370997 (51:7220)****[2]**S. M. Berman,*Excursions of stationary Gaussian processes above high moving boundaries*, Ann. Probability**1**(1973), 365-387. MR**0388514 (52:9350)****[3]**-,*Sojourns and extremes of Gaussian processes*, Ann. Probability**2**(1974), 999-1026. MR**0372976 (51:9178)****[4]**H. Cramé and M. R. Leadbetter,*Stationary and related stochastic processes*, Wiley, New York, 1967. MR**0217860 (36:949)****[5]**H. Crameé, 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**0290452 (44:7633)****[6]**X. Fernique,*Continuité des processus Gaussiens*, C. R. Acad. Sec. Paris Sér. A-B**258**(1964), 6058-6060. MR**0164365 (29:1662)****[7]**K. Itô and H. P. McKean,*Diffusion processes and their sample paths*, Springer, Berlin, 1974. MR**0345224 (49:9963)****[8]**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.**[9]**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.**[10]**C. Quails and H. Watanabe,*Asymptotic properties of Gaussian processes*, Ann. Inst. Math. Statist.**43**(1972), 580-596. MR**0307318 (46:6438)****[11]**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), 93-120. MR**0321204 (47:9737)****[12]**J. Pickands,*Upcrossing probabilities for stationary Gaussian processes*. Trans. Amer. Math. Soc.**145**(1969), 51-73. MR**0250367 (40:3606)****[13]**V. I. Piterbarg and V. P. Prisjazňjuk,*Asymptotic behavior of the probability of a large excursion for a nonstationary Gaussian process*, Teor. Verojatnost. i Mat. Statist.**18**(1978), 121-134. MR**0494458 (58:13319)****[14]**E. Seneta,*Regularly varying functions*, Lecture Notes in Math., vol. 508, Springer, Berlin, 1976. MR**0453936 (56:12189)****[15]**D. Siegmund,*Repeated significance tests for a normal mean*, Biometrika**64**(1977), 177-189. MR**0488546 (58:8076)****[16]**D. Slepian,*The one-sided barrier problem for Gaussian noise*, Bell System Tech. J.**41**(1962), 463-501. MR**0133183 (24:A3017)****[17]**M. J. Wichura,*Boundary crossing probabilities associated with Motoo's law of the iterated logarithm*, Ann. Probability**1**(1973), 437-456. MR**0373034 (51:9236)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
60G15,
60F10,
60J65

Retrieve articles in all journals with MSC: 60G15, 60F10, 60J65

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