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Appears in collection : Heavy Tails, Long-Range Dependence, and Beyond / Queues lourdes, dépendance de long terme et  au-delà

In this talk, we consider function-indexed normalized weighted integrated periodograms for equidistantly sampled multivariate continuous-time state space models which are multivariate continuous-time ARMA processes. Thereby, the sampling distance is fixed and the driving Lévy process has at least a finite fourth moment. Under different assumptions on the function space and the moments of the driving Lévy process we derive a central limit theorem for the function-indexed normalized weighted integrated periodogram. Either the assumption on the function space or the assumption on the existence of moments of the Lévy process is weaker. The results can be used to derive the asymptotic behavior of the Whittle estimator and to construct goodness-of-fit test statistics as the Grenander-Rosenblatt statistic and the Cramér-von Mises statistic.

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Citation data

  • DOI 10.24350/CIRM.V.19937703
  • Cite this video Fasen-Hartmann, Vicky (04/07/2022). Empirical spectral processes for stationary state space models. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19937703
  • URL https://dx.doi.org/10.24350/CIRM.V.19937703

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Bibliography

  • FASEN-HARTMANN, Vicky et MAYER, Celeste. Empirical spectral processes for stationary state space processes. arXiv preprint arXiv:2202.12589, 2022. - https://doi.org/10.48550/arXiv.2202.12589
  • BARDET, Jean‐Marc, DOUKHAN, Paul, et LEÓN, José Rafael. Uniform limit theorems for the integrated periodogram of weakly dependent time series and their applications to Whittle's estimate. Journal of Time Series Analysis, 2008, vol. 29, no 5, p. 906-945. - https://doi.org/10.1111/j.1467-9892.2008.00588.x
  • DAHLHAUS, Rainer. Empirical spectral processes and their applications to time series analysis. Stochastic Processes and their Applications, 1988, vol. 30, no 1, p. 69-83. - https://doi.org/10.1016/0304-4149(88)90076-2
  • DAHLHAUS, Rainer et POLONIK, Wolfgang. Empirical spectral processes for locally stationary time series. Bernoulli, 2009, vol. 15, no 1, p. 1-39. - http://dx.doi.org/10.3150/08-BEJ137
  • MIKOSCH, Thomas et NORVAIŠA, Rimas. Uniform convergence of the empirical spectral distribution function. Stochastic processes and their applications, 1997, vol. 70, no 1, p. 85-114. - https://doi.org/10.1016/S0304-4149(97)00053-7

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