### Abstract

Our main intention is to describe the behavior of the (cumulative) distribution function of the random variable M_{O,1}:=sup_{0≤s,t≤1} W(s, t) near O, where W denotes one-dimensional, two-parameter Brownian sheet. A remarkable result of Florit and Nualart asserts that M_{0,1} 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 M_{0,1} near O 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.

Original language | English |
---|---|

Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Stochastic Processes and their Applications |

Volume | 90 |

Issue number | 1 |

Publication status | Published - Nov 2000 |

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### Keywords

- Brownian sheet
- Quasi-sure analysis
- Tail probability

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Mathematics(all)
- Statistics and Probability

### Cite this

*Stochastic Processes and their Applications*,

*90*(1), 1-18.

**Boundary crossings and the distribution function of the maximum of Brownian sheet.** / Csáki, E.; Khoshnevisan, Davar; Shi, Zhan.

Research output: Contribution to journal › Article

*Stochastic Processes and their Applications*, vol. 90, no. 1, pp. 1-18.

}

TY - JOUR

T1 - Boundary crossings and the distribution function of the maximum of Brownian sheet

AU - Csáki, E.

AU - Khoshnevisan, Davar

AU - Shi, Zhan

PY - 2000/11

Y1 - 2000/11

N2 - Our main intention is to describe the behavior of the (cumulative) distribution function of the random variable MO,1:=sup0≤s,t≤1 W(s, t) near O, 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 O 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.

AB - Our main intention is to describe the behavior of the (cumulative) distribution function of the random variable MO,1:=sup0≤s,t≤1 W(s, t) near O, 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 O 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.

KW - Brownian sheet

KW - Quasi-sure analysis

KW - Tail probability

UR - http://www.scopus.com/inward/record.url?scp=0040044514&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040044514&partnerID=8YFLogxK

M3 - Article

VL - 90

SP - 1

EP - 18

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 1

ER -