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April, 1995 On the Distribution of Bubbles of the Brownian Sheet
Davar Khoshnevisan
Ann. Probab. 23(2): 786-805 (April, 1995). DOI: 10.1214/aop/1176988290


Let $W$ be a real-valued, two-parameter Brownian sheet. Let us define $N(t; h)$ to be the total number of bubbles of $W$ in $\lbrack 0, t\rbrack^2$, whose maximum height is greater than $h$. Evidently, $\lim_{h\downarrow 0} N(t; h) = \infty$ and $\lim_{t\uparrow\infty} N(t; h) = \infty$. It is the goal of this paper to provide fairly accurate estimates on $N(t; h)$ both as $t\rightarrow\infty$ and as $h\rightarrow 0$. Loosely speaking, we show that there are of order $h^{-3}$ many such bubbles as $h \downarrow 0$ and $t^3$ many, as $t\uparrow \infty$.


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Davar Khoshnevisan. "On the Distribution of Bubbles of the Brownian Sheet." Ann. Probab. 23 (2) 786 - 805, April, 1995.


Published: April, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0833.60044
MathSciNet: MR1334172
Digital Object Identifier: 10.1214/aop/1176988290

Primary: 60G17
Secondary: 60G15 , 60G60

Keywords: Brownian sheet , bubbles

Rights: Copyright © 1995 Institute of Mathematical Statistics

Vol.23 • No. 2 • April, 1995
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