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

For 0<?<1, let {Mathematical expression}. The questions addressed in this paper are motivated by a result due to Strassen: almost surely, lim sup_t?8U_?((t))=1-exp{-4(?^-1)-1}. We show that Strassen's result is closely related to a large deviations principle for the family of random variables U_?(t), t>0. Also, when ?=1, U_?(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 {Mathematical expression} as ??0, where D is a suitable discount function. These results also hold for symmetric random walks. © 1995 Plenum Publishing Corporation.