Open Access
August, 1973 Local Asymptotic Laws for Brownian Motion
N. C. Jain, S. J. Taylor
Ann. Probab. 1(4): 527-549 (August, 1973). DOI: 10.1214/aop/1176996884


Upper and lower functions are defined for the large values of $|X_d(t + u) - X_d(t - \nu)|$ as $(u + \nu) \downarrow 0$ where $X_d$ is a standard Brownian motion in $R^d$, and it is shown that the integral test for two-sided growth in $R^d$ is the same as that for one-sided growth in $R^{d+2}$. It is also shown that, for $d \geqq 4$, the lower asymptotic growth rate of $|X_d(t + u) - X_d(t - \nu)|$ for small $(u + \nu) = h$ is the same as the lower growth rate of $|X_{d-2}(t + h) - X_{d-2}(t)|$. Integral tests are also obtained for local asymptotic growth rates of the associated processes $P_d(a) = \inf_{t\geqq0} \{t: |X(t)| \geqq a\}$ and $M_d(t) = \sup_{0\leqq s\leqq t} |X_d(t)|$.


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N. C. Jain. S. J. Taylor. "Local Asymptotic Laws for Brownian Motion." Ann. Probab. 1 (4) 527 - 549, August, 1973.


Published: August, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0261.60053
MathSciNet: MR365732
Digital Object Identifier: 10.1214/aop/1176996884

Primary: 60J65
Secondary: 60G17

Keywords: absolute maximum process , Brownian motion , first passage time process , Integral test , Law of the iterated logarithm , local asymptotic laws , two-sided rate of escape , two-sided rate of growth , upper and lower classes

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 4 • August, 1973
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