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

E. Csáki, Davar Khoshnevisan, Zhan Shi

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1-18
Number of pages18
JournalStochastic Processes and their Applications
Volume90
Issue number1
Publication statusPublished - Nov 2000

Fingerprint

Boundary Crossing
Brownian Sheet
Density Function
Probability density function
Distribution functions
Two Parameters
Distribution Function
Reflection Principle
Brownian movement
Cumulative distribution function
Lebesgue Measure
Supremum
Random variables
Smooth function
Rectangle
Estimate
Brownian motion
Random variable
Denote
Analogue

Keywords

  • Brownian sheet
  • Quasi-sure analysis
  • Tail probability

ASJC Scopus subject areas

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

Cite this

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

In: Stochastic Processes and their Applications, Vol. 90, No. 1, 11.2000, p. 1-18.

Research output: Contribution to journalArticle

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