Laws of the iterated logarithm for \alpha-time Brownian motion

We introduce a class of iterated processes called $\alpha$-time Brownian motion for $0<\alpha \le 2$. 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 $$\underset{T\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}{T}^{-1\mathrm{/}2}(\log \log T)\underset{0\le t\le T}{\sup }\mathrm{\mid }{Z}_t\mathrm{\mid }={\pi }^2\sqrt{{\lambda }_1}a\mathrm{.}s\mathrm{.}$$ where ${\lambda }_1$ is the first eigenvalue for the Cauchy process in the interval $[-1,1]\mathrm{.}$ We also define the local time $L^{\ast }(x,t)$ and range $R^{*}(t)=|{x: Z(s)=x $\text{for some}$ s\leq t}|$fortheseprocessesfor$1<\alpha <2$. We prove that there are universal constants $c_R,c_L\in (0,\mathrm{\infty })$ such that $$\underset{t\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{R^{\ast }(t)}{(t\mathrm{/}\log \log t{)}^{1\mathrm{/}2\alpha }\log \log t}=c_{R}a\mathrm{.}s\mathrm{.}$$ $$\underset{t\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{\underset{x\in \text{RR}R}{\sup }{L}^{\ast }(x,t)}{(t\mathrm{/}\log \log t{)}^{1-1\mathrm{/}2\alpha }}=c_{L}a\mathrm{.}s\mathrm{.}$$