ISSUES CONCERNING THE APPROXIMATION UNDERLYING THE SPECTRAL REPRESENTATION THEOREMMarco Lippi () Econometric Theory, 2004, vol. 20, issue 02, pages 417-426 Abstract: 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 { (t), t T} is the limit in quadratic mean of a sequence of processes {S(n, t), t T}, each consisting of a finite sum of harmonic oscillations with stochastic weights. The natural issues, whether the approximation error (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 (t) S(n, t) is not stationary). Equivalently, when T is unbounded the number of harmonic oscillations necessary to approximate (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 (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). Date: 2004 Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: http://EconPapers.repec.org/RePEc:cup:etheor:v:20:y:2004:i:02:p:417-426_20 Access Statistics for this article More articles in Econometric Theory from Cambridge University Press |
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