The Annals of Probability

Local asymptotic classes for the successive primitives of Brownian motion

Aimé Lachal

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Abstract

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.

Article information

Source
Ann. Probab., Volume 25, Number 4 (1997), 1712-1734.

Dates
First available in Project Euclid: 7 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1023481108

Digital Object Identifier
doi:10.1214/aop/1023481108

Mathematical Reviews number (MathSciNet)
MR1487433

Zentralblatt MATH identifier
0903.60071

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 60F15: Strong theorems 60G15: Gaussian processes
Secondary: 60G17: Sample path properties 60J25

Keywords
Law of the iterated logarithm local asymptotic classes integral tests

Citation

Lachal, Aimé. Local asymptotic classes for the successive primitives of Brownian motion. Ann. Probab. 25 (1997), no. 4, 1712--1734. doi:10.1214/aop/1023481108. https://projecteuclid.org/euclid.aop/1023481108


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