Iterated Law of Iterated Logarithm

Abstract

Suppose $\varepsilon \in \lbrack 0,1)$ and let $\theta_\varepsilon(t) = (1 - \varepsilon)\sqrt{2t\ln_2 t}$. Let $L^\varepsilon_t$ denote the amount of local time spent by Brownian motion on the curve $\theta_\varepsilon(s)$ before time $t$. If $\varepsilon > 0$, then $\lim\sup_{t\rightarrow\infty} L^\varepsilon_t/\sqrt{2t\ln_2 t} = 2\varepsilon + o(\varepsilon)$. For $\varepsilon = 0$, a nontrivial lim sup is obtained when the normalizing function $\sqrt{2t\ln_2t}$ is replaced by $g(t) = \sqrt{t/\ln_2 t}\ln_3 t$.