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.

In the sampling and reconstruction of a random process a discrepancy may occur between its multiplicity and that of its reconstructed version. It is shown that under general conditions an error due to this discrepancy is not present in the total error due to sampling and reconstruction. The problem is studied in the more general framework of tracking a random process of arbitrary multiplicity with one of unit multiplicity; there too, the tracking error can be made arbitrarily small.

The problem of determining a shaping filter for nonstationary colored noise is considered. The shaping filter transforms white noise into a possibly nonstationary random process (having no white noise component) with a specified covariance function. A set of conditions to be satisfied by the covariance function leads to the determination of a shaping filter. The shaping filter coefficients are simply related to the solution of a matrix Riccati equation. In order to formulate the Riccati equation, basic results concerning the mean-square differentiability of a random process are developed. If the Riccati equation can not be defined, an autonomous (zero-input) shaping filter may be easily determined.