Then, the process can be approximated as closely as desired by restricting the index to a set dense in the indexing set. Stretching mathematical rigor further, a random process can be represented by its values at a discrete set of points in its domain of definition. It is then clear how the foregoing discussion for random variables can be extended to the case of random processes. In the context of the finite element analysis of a given system, the random processes involved are substituted for by random variables that are so chosen as to coincide with some local average of the process over each element. It is expected, however, that the result would depend to a notable extent on the averaging method used.
Local averages are usually of two kinds.
It is clear that, in general, the first approach smoothes the random process, whereas the second approach introduces additional irregularities. This suggests that the two approaches provide lower and upper bounds, respectively, for the random behavior of the process. In the limit, as the size of each subset becomes vanishingly small, the representation resulting from the two approaches should converge to the exact process. It is obvious that a relatively large number of random variables is required to represent a random process in this fashion. It is noted that local averaging parallels pointwise approximations of deterministic functions.