@article{EJP327,
author = {Erkan Nane},
title = {Laws of the iterated logarithm for α-time Brownian motion},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {11},
year = {2006},
keywords = {Brownian motion, symmetric \$alpha\$-stable process, \$alpha\$-time Brownian motion, local time, Chung's law, Kesten's law.},
abstract = {We introduce a class of iterated processes called $\alpha$-time Brownian motion for $0<\alpha \leq 2$. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric $\alpha$-stable process. We prove a Chung-type law of the iterated logarithm (LIL) for these processes which is a generalization of LIL proved in cite{hu} for iterated Brownian motion. When $\alpha =1$ it takes the following form $$ \liminf_{T\to\infty}\ T^{-1/2}(\log\log T) \sup_{0\leq t\leq T}|Z_{t}|=\pi^{2}\sqrt{\lambda_{1}} \quad a.s. $$ where $\lambda_{1}$ is the first eigenvalue for the Cauchy process in the interval $[-1,1].$ We also define the local time $L^{*}(x,t)$ and range $R^{*}(t)=|{x: Z(s)=x \text{ for some } s\leq t}|$ for these processes for $1<\alpha <2$. We prove that there are universal constants $c_{R},c_{L}\in (0,\infty) $ such that $$ \limsup_{t\to\infty}\frac{R^{*}(t)}{(t/\log \log t)^{1/2\alpha}\log \log t}= c_{R} \quad a.s. $$ $$ \liminf_{t\to\infty} \frac{\sup_{x\in {R}}L^{*}(x,t)}{(t/\log \log t)^{1-1/2\alpha}}= c_{L} \quad a.s. $$},
pages = {no. 18, 434-459},
issn = {1083-6489},
doi = {10.1214/EJP.v11-327},
url = {http://ejp.ejpecp.org/article/view/327}}