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

Author

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  • Csáki, Endre
  • Khoshnevisan, Davar
  • Shi, Zhan

Abstract

Our main intention is to describe the behavior of the (cumulative) distribution function of the random variable M0,1 := sup0[less-than-or-equals, slant]s,t[less-than-or-equals, slant]1 W(s,t) near 0, where W denotes one-dimensional, two-parameter Brownian sheet. A remarkable result of Florit and Nualart asserts that M0,1 has a smooth density function with respect to Lebesgue's measure (cf. Florit and Nualart, 1995. Statist. Probab. Lett. 22, 25-31). Our estimates, in turn, seem to imply that the behavior of the density function of M0,1 near 0 is quite exotic and, in particular, there is no clear-cut notion of a two-parameter reflection principle. We also consider the supremum of Brownian sheet over rectangles that are away from the origin. We apply our estimates to get an infinite-dimensional analogue of Hirsch's theorem for Brownian motion.

Suggested Citation

  • Csáki, Endre & Khoshnevisan, Davar & Shi, Zhan, 2000. "Boundary crossings and the distribution function of the maximum of Brownian sheet," Stochastic Processes and their Applications, Elsevier, vol. 90(1), pages 1-18, November.
    • Handle: RePEc:eee:spapps:v:90:y:2000:i:1:p:1-18
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    References listed on IDEAS

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    1. Florit, Carme & Nualart, David, 1995. "A local criterion for smoothness of densities and application to the supremum of the Brownian sheet," Statistics & Probability Letters, Elsevier, vol. 22(1), pages 25-31, January.
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    Cited by:

    1. Enkelejd Hashorva, 2010. "Boundary Non-crossings of Brownian Pillow," Journal of Theoretical Probability, Springer, vol. 23(1), pages 193-208, March.
    2. Molchan, G., 2008. "Unilateral small deviations of processes related to the fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 118(11), pages 2085-2097, November.

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