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

Endre Csáki, Davar Khoshnevisan and Zhan Shi

Stochastic Processes and their Applications, 2000, vol. 90, issue 1, 1-18

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.

Keywords: Tail; probability; Quasi-sure; analysis; Brownian; sheet (search for similar items in EconPapers)
Date: 2000
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Citations: View citations in EconPapers (2)

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