We prove a small deviation theorem of a new form for the functional central limit theorem for partial sums of independent, identically distributed finite-dimensional random vectors. The result is applied to obtain a functional form of the Chung-Jain-Pruitt law of the iterated logarithm which is also a strong speed of convergence theorem refining Strassen's invariance principle.
"Small Deviations in the Functional Central Limit Theorem with Applications to Functional Laws of the Iterated Logarithm." Ann. Probab. 11 (1) 78 - 101, February, 1983. https://doi.org/10.1214/aop/1176993661