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

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.