Issues concerning the approximation underlying the spectral representation theorem

In many important textbooks the formal statement of the spectral representation theorem is followed by a process version, usually informal, stating that any stationary stochastic process {xi(t), t is an element of T} is the limit in quadratic mean of a sequence of processes {S(n, t), t is an element of T}, each consisting of a finite sum of harmonic oscillations with stochastic weights. The natural issues, whether the approximation error xi(t) - S(n,t) is stationary or whether at least it converges to zero uniformly in t, have not been explicitly addressed in the literature. The paper shows that in all relevant cases, for T unbounded the process convergence is not uniform in t (so that xi(t) - S(n, t) is not stationary). Equivalently, when T is unbounded the number of harmonic oscillations necessary to approximate xi(t) with a preassigned accuracy depends on t. The conclusion is that the process version of the spectral representation theorem should explicitly mention that in general the approximation of xi(t) by a finite sum of harmonic oscillations, given the accuracy, is valid for t belonging to a bounded subset of the real axis (of the set of integers in the discrete-parameter case).