Local asymptotic classes for the successive primitives of Brownian motion

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.