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Uniform limit theorems for the integrated periodogram of weakly dependent time series and their applications to Whittle's estimate

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  • Jean‐Marc Bardet
  • Paul Doukhan
  • José Rafael León

Abstract

. We prove uniform convergence results for the integrated periodogram of a weakly dependent time series, namely a strong law of large numbers and a central limit theorem. These results are applied to Whittle's parametric estimation. Under general weak‐dependence assumptions, the strong consistency and asymptotic normality of Whittle's estimate are established for a large class of models. For instance, the causal θ‐weak dependence property allows a new and unified proof of those results for autoregressive conditionally heteroscedastic (ARCH)(∞) and bilinear processes. Non‐causal η‐weak dependence yields the same limit theorems for two‐sided linear (with dependent inputs) or Volterra processes.

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  • Jean‐Marc Bardet & Paul Doukhan & José Rafael León, 2008. "Uniform limit theorems for the integrated periodogram of weakly dependent time series and their applications to Whittle's estimate," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(5), pages 906-945, September.
    • Handle: RePEc:bla:jtsera:v:29:y:2008:i:5:p:906-945
    DOI: 10.1111/j.1467-9892.2008.00588.x
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    1. Vicky Fasen-Hartmann & Celeste Mayer, 2022. "Whittle estimation for continuous-time stationary state space models with finite second moments," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(2), pages 233-270, April.
    2. Fasen-Hartmann, Vicky & Mayer, Celeste, 2023. "Empirical spectral processes for stationary state space models," Stochastic Processes and their Applications, Elsevier, vol. 155(C), pages 319-354.
    3. Sourav Das & Suhasini Subba Rao & Junho Yang, 2021. "Spectral methods for small sample time series: A complete periodogram approach," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(5-6), pages 597-621, September.
    4. Tata Subba Rao & Granville Tunnicliffe Wilson & Joao Jesus & Richard E. Chandler, 2017. "Inference with the Whittle Likelihood: A Tractable Approach Using Estimating Functions," Journal of Time Series Analysis, Wiley Blackwell, vol. 38(2), pages 204-224, March.

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