## Laws of the iterated logarithm for $$\alpha$$-time Brownian motion.(English)Zbl 1121.60085

Summary: We introduce a class of iterated processes called $$\alpha$$-time Brownian motion for $$0<\alpha<2$$. 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 by Y. Hu, D. Pierre-Loti-Viaud and Z. Shi [J. Theor. Probab. 8, No. 2, 303–319 (1995; Zbl 0816.60027)] for iterated Brownian motion. When $$\alpha= 1$$ it takes the following form $\liminf_{T\to\infty}T^{-1/2}(\log\log T)\sup_{0 \leq t\leq T}|Z_t|=\pi^2\sqrt{\lambda_1}\text{ a.s.}$ 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 $$R^*(t)=|\{x:Z(s)=x$$ 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 $\limsup_{t\to\infty}\frac{R^*(t)} {(t/\log\log t)^{1/2\alpha}\log\log t}=c_R\text{ a.s.}, \quad \liminf_{t\to\infty} \frac{\sup_{x\in\mathbb{R}} L^*(x,t)}{(t/\log\log t)^{1-1/2\alpha}}=c_L\text{ a.s.}$

### MSC:

 60J65 Brownian motion 60K99 Special processes

Zbl 0816.60027
Full Text: