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

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

2007-01-25

arxiv.org/abs/math/0701739v1

Journal of Time Series Analysis (2008) \`a para\^itre

Mathematics

Statistics Theory

Scientific paper

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.

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