language: en

hal-00126489, version 1 - 25 Jan 2007

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

Jean-Marc Bardet a* , Paul Doukhan ab , José Rafael Léon c?
a SAMOS-MATISSE-CES, Université e Paris I,
90 rue de Tolbiac, 75013 Paris, FRANCE
bardet@univ-paris1.fr
b LS-CREST,
Timbre J340, 3 avenue Pierre Larousse, 92240 Malakoff, FRANCE
doukhan@ensae.fr
c Universidad Central de Venezuela, Escuela de Matemática.
Ap. 47197 Los Chaguaramos, Caracas 1041-A VENEZUELA
jleon@euler.ciens.ucv.ve

Abstract 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 ?-weak dependence property allows a new and unified proof of those results for ARCH(?) and bilinear processes. Non causal ?-weak dependence yields the same limit theorems for two-sided linear (with dependent inputs) or Volterra processes.

Key words: Periodogram, Weak dependence, Whittle estimate. AMS Subject Classification: 60F17, 60F25, 62M09, 62M10, 62M15.

[Running title: Uniform limit theorems for the integrated periodogram and Whittle's estimate.]

1 Introduction