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 $M_d(t) = \sup_{0\leqq s\leqq t} |X_d(t)|$.
Publié le : 1973-08-14
Classification:
Brownian motion,
local asymptotic laws,
law of the iterated logarithm,
two-sided rate of growth,
two-sided rate of escape,
first passage time process,
absolute maximum process,
upper and lower classes,
integral test,
60J65,
60G17
@article{1176996884,
author = {Jain, N. C. and Taylor, S. J.},
title = {Local Asymptotic Laws for Brownian Motion},
journal = {Ann. Probab.},
volume = {1},
number = {5},
year = {1973},
pages = { 527-549},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996884}
}
Jain, N. C.; Taylor, S. J. Local Asymptotic Laws for Brownian Motion. Ann. Probab., Tome 1 (1973) no. 5, pp. 527-549. http://gdmltest.u-ga.fr/item/1176996884/