On the iterated logarithm law for local time

Abstract:

If $s(t,x)$ is the local time of a Brownian motion, we show that $\theta (\alpha ) = \lim {\sup _{t \to \infty }}{\inf _{\left | x \right | \leqslant \alpha {t^{1/2}}{{(2\log \log t)}^{ - 1/2}}}}s(t,x){(2t\log \log t)^{ - 1/2}}$ satisfies \[ {((1 - {\alpha ^{1/2}}) \vee 0)^2} \leqslant \theta (\alpha ) \leqslant {(2\alpha )^{ - 1}} \wedge 1.\] In particular, it follows from a result of Kesten that \[ \lim \sup \limits _{t \to \infty } s(t,x){(2t\log \log t)^{ - 1/2}} = 1\] for all $x$ a.s.