Uniform Limit Theorems for the Integrated Periodogram of Weakly Dependent Time Series and their Applications to Whittle’s Estimate

We prove uniform convergence results like a law of large numbers and a central limit theoremfor the integrated periodogram of a weak dependent time series. Those probabilistic results areused for Whittle’s parametric estimation. Using a general weakly dependent frame, we derivenew results, i.e. uniform limit theorems and asymptotic normality of the Whittle estimate, fora large variety of models. For instance, the causal -weak dependence property allows a newand unique proof of those results for LARCH(1) and bilinear processes, whereas the non causal-weak dependence property provides for the first time those limit theorems for two sided linear,Volterra, bilinear and LARCH(1) processes.