Boundary crossings and the distribution function of the maximum of Brownian sheet

Stochastic Processes and their Applications

Volumn 90, Issue 1, 2000, Pages 1-18

  Csáki, Endre ^a   Khoshnevisan, Davar ^b   Shi, Zhan ^c  

^a INSTITUTE OF EXPERIMENTAL MEDICINE   (Hungary)

^b UNIVERSITY OF UTAH   (United States)

^c University of Paris VII LPSM partially supported by ERC Starting Grant 680275 MALIG   (France)

Author keywords

Brownian sheet; Quasi sure analysis; Tail probability

Indexed keywords

EID: 0040044514     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-4149(00)00031-4     Document Type: Article

References (33)

1. Adler, R.J., 1981. The Geometry of Random Fields. Wiley, Chichester.
2. Adler, R.J., 1984. The supremum of a particular Gaussian field. Ann. Probab. 12, 436-444.
3. Adler, R.J., 1990. Continuity, Extrema and Related Topics for General Gaussian Processes. IMS Lecture Notes-Monograph Series, Vol. 12. Institute of Mathematical Statistics, Hay ward, CA.
4. Bañuelos, R., Smits, R.G., 1997. Brownian motion in cones. Probab. Theory Related Fields 108, 299-319.
5. Berman, S.M., 1991. Sojourns and Extremes of Stochastic Processes. Wadsworth and Brooks/Cole, Belmont, CA.
6. Bramson, M., Griffeath, D., 1991. Capture problems for coupled random walks. In: Durrett, R., Kesten, H. (Eds.), Random Walks, Brownian Motion and Interacting Particle Systems. Birkhäuser, Boston, pp. 153-188.
7. Cabaña, E.M., Wschebor, M., 1982. The two parameter Brownian bridge - Kolmogorov inequalities and upper and lower bounds for the distribution of the maximum. Ann. Probab. 10, 289-302.
8. Csáki, E., 1978. On the lower limits of maxima and minima of Wiener process and partial sums. Z. Wahrsch. Verw. Gebiete 43, 205-221.
9. Dalang, R., Walsh, J.B., 1993. Geography of the level sets of the Brownian sheet. Probab. Theory Related Fields 96, 153-176.
10. Dembo, A., Zeitouni, O., 1998. Large Deviations: Techniques and Applications, 2nd Edition. Springer, New York.
11. Feller, W., 1957. An Introduction to Probability Theory and its Applications, Vol. I, 2nd Edition. Wiley, New York.
12. Feller, W., 1971. An Introduction to Probability Theory and its Applications, Vol. II, 2nd Edition. Wiley, New York.
13. Florit, C., Nualart, D., 1995. A local criterion for smoothness of densities and application to the supremum of the Brownian sheet. Statist. Probab. Lett. 22, 25-31.
14. Fukushima, M., 1984. Basic properties of Brownian motion and a capacity on the Wiener space. J. Math. Soc. Japan 36, 161-176.
15. Goodman, V., 1975. Distribution estimates for functionals of two-parameter Wiener process. Ann. Probab. 4, 977-982.
16. Hirsch, W.M., 1965. A strong law for the maximum cumulative sum of independent random variables. Comm. Pure Appl. Math. 18, 109-127.
17. Hogan, M.L., Siegmund, D., 1986. Large deviations for the maxima of some random fields. Adv. Appl. Math. 7, 2-22.
18. Kahane, J.-P., 1985. Some Random Series of Functions, 2nd Edition. Cambridge University Press, Cambridge.
19. Kendall, W.S., 1980. Contours of Brownian processes with several-dimensional time. Z. Wahrsch. Verw. Gebiete 52, 267-276.
20. Kesten, H., 1992. An absorption problem for several Brownian motions. Seminar on Stochastic Process 91, Birkhäuser, Boston, pp. 59-72.
21. Khoshnevisan, D., Shi, Z., 1999. Brownian sheet and capacity. Ann. Probab. 27, 1135-1159.
22. Lawler, G.F., 1991. Intersections of Random Walks. Birkhäuser, Boston.
23. Ledoux, M., Talagrand, M., 1991. Probability in Banach Spaces. Springer, Berlin.
24. Malliavin, P., 1978. Stochastic calculus of variation and hypoelliptic operators. Proceedings of the International Symposium on Stochastic and Differential Equations, Kyoto, 1976. Wiley, New York, pp. 195-263.
25. Orey, S., Pruitt, W.E., 1973. Sample functions of the N-parameter Wiener process. Ann. Probab. 1, 138-163.
26. Pemantle, R., Peres, Y., 1995. Critical random walk in random environment on trees. Ann. Probab. 23, 105-140.
27. Siegmund, D., 1988. Approximate tail probabilities for the maxima of some random fields. Ann. Probab. 16, 487-501.
28. Slepian, D., 1962. The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41, 463-501.
29. Spitzer, F., 1958. Some theorems concerning 2-dimensional Brownian motion. Trans. Amer. Math. Soc. 87, 187-197.
30. Talagrand, M., 1994. The small ball problem for Brownian sheet. Ann. Probab. 22, 1331-1354.
31. Williams, D., 1982. Appendix to P.-A. Meyer: Note sur les processus d'Ornstein-Uhlenbeck. Sém. de Probab. XVI. Lecture Notes in Mathematics, Vol. 920, Springer, Berlin, p. 133.
32. Worsley, K.J., 1995a. Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images. Ann. Statist. 23, 640-669.
33. Worsley, K.J., 1995b. Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics. Adv. Appl. Probab. 27, 943-959.