An independent stationary process {Xi}∞i=1 in ℝn is perturbed by a sequence of Euclidean motions to obtain a new process {Yi}∞i=1. Criteria are given for the singularity or equivalence of these processes. When the distribution of the X process has finite Fisher information, the criteria are necessary and sufficient. Moreover, it is proved that it is exactly under the condition of finite Fisher information that the criteria are necessary and sufficient.