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

Jean‐Marc Bardet, Paul Doukhan and José Rafael León

Journal of Time Series Analysis, 2008, vol. 29, issue 5, 906-945

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.

Date: 2008
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https://doi.org/10.1111/j.1467-9892.2008.00588.x

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