On the rate of convergence in Strassen's law of the iterated logarithm

Abstract

Let W(t) be a standard Wiener process and let L be the compact class figuring in Strassen's law of the iterated logarithm. We investigate the rate of convergence to zero of the variable $$\mathop {inf}\limits_{f \in \mathfrak{L}} {\text{ }}\mathop {{\text{sup}}}\limits_{{\text{0}} \leqq x \leqq 1} {\text{ |}}W(xT)(2T log log T)^{ - \frac{1}{2}} - f(x)|.$$ It is shown that as T→∞, (log log T)-α belongs to the upper class of this variable if α 2/3.