Spectral characteristics of Harmonizable VARMA processes

Dominique Dehay, Anna E. Dudek and Jean‐Marc Freyermuth

Scandinavian Journal of Statistics, 2026, vol. 53, issue 2, 613-647

Abstract: Harmonizable processes form a wide class of nonstationary processes, which admit a convenient Fourier analysis and have spectral distributions characterized by correlated components. They have been proven to be useful in many fields of application, e.g., in communication, seismology, EEG data analysis, etc. In this paper, we introduce a parametric form for harmonizable processes, namely Harmonizable Vector AutoRegressive and Moving Average models (HVARMA). In the same spirit as standard VARMA models, they are derived as a unique solution of a difference equation based on a properly defined concept of harmonizable noise. We exhibit their spectral characteristics and derive results for least‐squares parameter estimation in a fundamental case. We notably obtain, in some particular cases of the explosive regime, unusual asymptotic laws. Most importantly, we provide an effective way to generate realizations from those novel processes. Our modeling choice for harmonizable noise induces second‐order stationary dependencies among the components of the spectral distribution of the harmonizable time series. We choose to model them using a periodic stationary VARMA(p′,q′)$$ \left({p}^{\prime },{q}^{\prime}\right) $$ process, resulting in the so‐called HVARMA(p,q)$$ \left(p,q\right) $$–(p′,q′)$$ \left({p}^{\prime },{q}^{\prime}\right) $$ model. We characterize its spectral properties and illustrate its ability to capture a vast range of nonstationary behaviors through examples of realizations using various models.

Date: 2026
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