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It is shown that any random process with finite energy can be approximated arbitrarily closely by a mean-square continuous process. We obtain approximants that in addition have continuous sample paths with probability 1. By a straightforward extension of these results, the approximating process can be made to have differentiable sample paths. The approximation can be performed in real time, in the sense that the approximating process constructed can be regarded as the output of a causal linear time-invariant system whose input is the finite energy process that is to be approximated.