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 \[ \limsup_{t\to\infty}\frac{R^*(t)} {(t/\log\log t)^{1/2\alpha}\log\log t}=c_R\text{ a.s.}, \quad \liminf_{t\to\infty} \frac{\sup_{x\in\mathbb{R}} L^*(x,t)}{(t/\log\log t)^{1-1/2\alpha}}=c_L\text{ a.s.} \]


60J65 Brownian motion
60K99 Special processes


Zbl 0816.60027
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