Some asymptotic results related to the law of iterated logarithm for Brownian motion

Abstract

For 0<γ<1, let$U_\gamma (t) = t^{ - 1} \int_0^t {1_{\{ B_3 > \sqrt {\gamma s\log \log s} \} } } ds$. The questions addressed in this paper are motivated by a result due to Strassen: almost surely, lim sup _t→∞ U _γ((t))=1−exp{−4(γ⁻¹)−1}. We show that Strassen's result is closely related to a large deviations principle for the family of random variablesU _γ (t), t>0. Also, when γ=1,U _γ (t)→0 almost surely and we obtain some bounds on the rate of convergence. Finally, we prove an analogous limit theorem for discounted averages of the form$\lambda \smallint _0^\infty D(\lambda s) 1_{(B_3 \sqrt {2\gamma s \log \log s} )} ds$ as λ↓0, whereD is a suitable discount function. These results also hold for symmetric random walks.