Laws of the iterated logarithm for α-time Brownian motion

Abstract

We introduce a class of iterated processes called $α$-time Brownian motion for $0<α\leq 2$. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric $α$-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 $α=1$ it takes the following form $$ \liminf_{T\to\infty}T^{-1/2}(\log \log T) \sup_{0\leq t\leq T}|Z_{t}|=π^{2}\sqrt{λ_{1}} a.s. $$ where $λ_{1}$ 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 $1<α<2$. We prove that there are universal constants $c_{R},c_{L}\in (0,\infty) $ such that $$ \limsup_{t\to\infty}\frac{R^{*}(t)}{(t/\log \log t)^{1/2α}\log \log t}= c_{R} a.s. $$ $$ \liminf_{t\to\infty} \frac{\sup_{x\in \RR{R}}L^{*}(x,t)}{(t/\log \log t)^{1-1/2α}}= c_{L} a.s. $$