Uniform limit theorems for the integrated periodogram of weakly dependent time series and their applications to Whittle's estimate |
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Authors: | Jean‐Marc Bardet Paul Doukhan José Rafael León |
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Affiliation: | 1. Université Paris I;2. LS‐CREST;3. Universidad Central de Venezuela |
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Abstract: | 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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Keywords: | Periodogram weak dependence Whittle estimate 60F17 60F25 62M09 62M10 62M15 |
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