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Continuous-time locally stationary time series models

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  20 June 2023

Annemarie Bitter ,
Robert Stelzer  and
Bennet Ströh
Annemarie Bitter*
Affiliation:
Ulm University
Robert Stelzer*
Affiliation:
Ulm University
Bennet Ströh*
Affiliation:
Imperial College London
*
*Postal address: Helmholtzstraße 18, 89069 Ulm, Germany.
*Postal address: Helmholtzstraße 18, 89069 Ulm, Germany.
****Postal address: 180 Queen’s Gate, London, SW7 2AZ, UK. Email address: b.stroh@imperial.ac.uk
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Abstract

We adapt the classical definition of locally stationary processes in discrete time (see e.g. Dahlhaus, ‘Locally stationary processes’, in Time Series Analysis: Methods and Applications (2012)) to the continuous-time setting and obtain equivalent representations in the time and frequency domains. From this, a unique time-varying spectral density is derived using the Wigner–Ville spectrum. As an example, we investigate time-varying Lévy-driven state space processes, including the class of time-varying Lévy-driven CARMA processes. First, the connection between these two classes of processes is examined. Considering a sequence of time-varying Lévy-driven state space processes, we then give sufficient conditions on the coefficient functions that ensure local stationarity with respect to the given definition.

Keywords

Non-stationary processesLévy-driven state space modelsCARMA processesspectral densityWigner–Ville spectrum

MSC classification

Primary: 60G07: General theory of processes 60G51: Processes with independent increments; L'evy processes
Secondary: 62M15: Spectral analysis
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 3 , September 2023 , pp. 965 - 998
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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