Spectral analysis of multivariate stationary Hawkes processes
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915933298098176 |
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| author | Tang, Yifu Kresin, Conor Baeumer, Boris Wang, Ting |
| author_facet | Tang, Yifu Kresin, Conor Baeumer, Boris Wang, Ting |
| contents | We establish the asymptotic validity of frequency-domain inference for stationary multivariate Hawkes processes under mild conditions, bridging the gap between theory and application. By developing upper-bounds on the reduced cumulant measures from the cluster representation of the Hawkes processes, we prove a functional central limit theorem and, as a consequence, consistency of the Whittle estimator under stationarity alone (i.e., the spectral radius of the interactions matrix $ρ(\boldsymbolν)<1$), applicable to Hawkes processes with heavy-tailed mutual-excitation kernels. Under mild extra moment conditions, we further obtain asymptotic normality with an explicit limiting covariance in terms of second- and fourth-order cumulant spectral densities. We also propose a simple frequency-domain method to detect joint independence of subprocesses of a multivariate Hawkes process. The performance of the Whittle estimator and the test of independence are demonstrated via simulation studies. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_10376 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral analysis of multivariate stationary Hawkes processes Tang, Yifu Kresin, Conor Baeumer, Boris Wang, Ting Statistics Theory We establish the asymptotic validity of frequency-domain inference for stationary multivariate Hawkes processes under mild conditions, bridging the gap between theory and application. By developing upper-bounds on the reduced cumulant measures from the cluster representation of the Hawkes processes, we prove a functional central limit theorem and, as a consequence, consistency of the Whittle estimator under stationarity alone (i.e., the spectral radius of the interactions matrix $ρ(\boldsymbolν)<1$), applicable to Hawkes processes with heavy-tailed mutual-excitation kernels. Under mild extra moment conditions, we further obtain asymptotic normality with an explicit limiting covariance in terms of second- and fourth-order cumulant spectral densities. We also propose a simple frequency-domain method to detect joint independence of subprocesses of a multivariate Hawkes process. The performance of the Whittle estimator and the test of independence are demonstrated via simulation studies. |
| title | Spectral analysis of multivariate stationary Hawkes processes |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2604.10376 |