In this paper, we investigate decompositions, dilations and compressions of discrete-time stochastic processes from the viewpoint of second-order moments. We first state a general decomposition theorem that plays an important role throughout this paper. Then we investigate several types of decompositions together with dilations and compressions as special cases of this theorem. In particular, decompositions into two processes are of special interest. For example, it is shown that under a mild condition, any -valued square-integrable discrete-time stochastic process with mean 0 can be expressed as the sum of two weak white noise processes with mean 0. We also propose a concrete decomposition algorithm for weakly stationary time series. Although we will not go into the details of the practical aspects, the obtained results have possible applications to time series analysis and secret sharing schemes. These will be closely examined in a separate paper.