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Issues Concerning The Approximation Underlying The Spectral Representation Theorem

Listed author(s):
  • Lippi, Marco

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).

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Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 20 (2004)
Issue (Month): 02 (April)
Pages: 417-426

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Handle: RePEc:cup:etheor:v:20:y:2004:i:02:p:417-426_20
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