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October 1997 Local asymptotic classes for the successive primitives of Brownian motion
Aimé Lachal
Ann. Probab. 25(4): 1712-1734 (October 1997). DOI: 10.1214/aop/1023481108


Let $(B(t))_{t \geq 0}$ be the linear Brownian motion starting at 0, and set $X_n (t) =(1/n!)\int_0^t(t-s)^s dB(s)$. Watanabe stated a law of the iterated logarithm for the process $(X_1(t))_{t \geq 0}$, among other things. This paper proposes an elementary proof of this fact, which can be extended to the general case $n\geq 1$. Next, we study the local asymptotic classes (upper and lower) of the $(n +1)$ -dimensional process $U_n =(B, X_1,\ldots,X_n)$ near zero and infinity, and the results obtained are extended to the case where $B$ is the $d$-dimensional Brownian motion.


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Aimé Lachal. "Local asymptotic classes for the successive primitives of Brownian motion." Ann. Probab. 25 (4) 1712 - 1734, October 1997.


Published: October 1997
First available in Project Euclid: 7 June 2002

zbMATH: 0903.60071
MathSciNet: MR1487433
Digital Object Identifier: 10.1214/aop/1023481108

Primary: 60F15 , 60G15 , 60J65
Secondary: 60G17 , 60J25

Keywords: integral tests , Law of the iterated logarithm , local asymptotic classes

Rights: Copyright © 1997 Institute of Mathematical Statistics

Vol.25 • No. 4 • October 1997
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