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 law of large numbers and a central limit theorem. These results are applied to Whittle's parametric estimation. Under general weak-dependence assumptions we derive uniform limit theorems and asymptotic normality of Whittle's estimate for a large class of models. For instance the causal $\theta$-weak dependence property allows a new and unified proof of those results for ARCH($\infty$) and bilinear processes. Non causal $\eta$-weak dependence yields the same limit theorems for two-sided linear (with dependent inputs) or Volterra processes.