A relationship between the path structure of two real discrete time stochastic processes is deduced from inequalities between their transition functions. The approach is to define processes equivalent to the two on a common space so that pointwise inequalities are possible. An iterated logarithm type law for random walks is given as a particular application of the general method.
"The Comparison Method for Stochastic Processes." Ann. Probab. 3 (1) 80 - 88, February, 1975. https://doi.org/10.1214/aop/1176996450