Decisions in Economics and Finance

, Volume 39, Issue 2, pp 203–234 | Cite as

Isometric operators on Hilbert spaces and Wold decomposition of stationary time series

  • Federico SeverinoEmail author


Wold Theorem plays a fundamental role in the decomposition of weakly stationary time series. It provides a moving average representation of the process under consideration in terms of uncorrelated innovations, whatever the nature of the process is. From an empirical point of view, this result enables to identify orthogonal shocks, for instance in macroeconomic and financial time series. More theoretically, the decomposition of weakly stationary stochastic processes can be seen as a special case of the Abstract Wold Theorem, that allows to decompose Hilbert spaces by using isometric operators. In this work we explain this link in detail, employing the Hilbert space spanned by a weakly stationary time series and the lag operator as isometry. In particular, we characterize the innovation subspace by exploiting the adjoint operator. We also show that the isometry of the lag operator is equivalent to weak stationarity. Our methodology, fully based on operator theory, provides novel tools useful to discover new Wold-type decompositions of stochastic processes, in which the involved isometry is no more the lag operator. In such decompositions the orthogonality of innovations is ensured by construction since they are derived from the Abstract Wold Theorem.


Isometry Hilbert space Wold decomposition Stationary time series 

JEL Classification

C18 C22 



We wish to thank Fulvio Ortu, Andrea Tamoni and Claudio Tebaldi for fruitful discussions on the topic.


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Copyright information

© Springer-Verlag Italia 2016

Authors and Affiliations

  1. 1.Department of FinanceBocconi UniversityMilanItaly

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