 Issues Concerning the Approximation Underlying the Spectral Representation TheoremLEM Papers Series from Laboratory of Economics and Management (LEM), Sant'Anna School of Advanced Studies, Pisa, Italy Abstract: In many important textbooks the formal statement of the Spectral RepresentationTheorem is followed by a process version, usually informal, stating thatany stationary stochastic process g is the limit in quadratic mean of asequence of processes, each consisting of a finite sum of harmonicoscillations with stochastic weights. The natural issues, whether the approximationerror is stationary, or whether at least it converges to zero uniformly int , have not been explicitly addressed in the literature. The paper shows that in allrelevant cases, for T unbounded the process convergence is not uniform in t. Equivalently, when T is unbounded the numberof harmonic oscillations necessary to approximate a stationary stochastic process with a preassigned accuracydepends on t . The conclusion is that the process version of the Spectral RepresentationTheorem should explicitely mention that in general the approximation of a stationary stochastic processby a finite sum of harmonic oscillations, given the accuracy, is valid for t belongingto a bounded subset of the real axis (of the set of integers in the discrete-parametercase). Keywords: Stochastic; processes.; Stationarity.; Spectral; analysis. (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:ssa:lemwps:2003/14 Access Statistics for this paper More papers in LEM Papers Series from Laboratory of Economics and Management (LEM), Sant'Anna School of Advanced Studies, Pisa, Italy Contact information at EDIRC. |
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