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Issues Concerning the Approximation Underlying the Spectral Representation Theorem

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  • Marco Lippi

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).

Suggested Citation

  • Marco Lippi, 2003. "Issues Concerning the Approximation Underlying the Spectral Representation Theorem," LEM Papers Series 2003/14, Laboratory of Economics and Management (LEM), Sant'Anna School of Advanced Studies, Pisa, Italy.
  • Handle: RePEc:ssa:lemwps:2003/14
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    Keywords

    Stochastic processes. Stationarity. Spectral analysis.;

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