For processes with independent increments and multi-dimensional time parameters, analogues of Strassen's version of the law of the iterated logarithm are proven using standard large-deviation and truncation techniques. Applications to empirical processes are included.
"Some Strassen-Type Laws of the Iterated Logarithm for Multiparameter Stochastic Processes with Independent Increments." Ann. Probab. 1 (2) 272 - 296, April, 1973. https://doi.org/10.1214/aop/1176996980