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Théorèmes limites fonctionnels et estimation de la densité spectrale pour des suites stationnaires.

Abstract : This works aims at deriving asymptotic results for some distances between the distribution function and the corresponding empirical distribution function of a stationary sequences (Cramér-Von Mises distance or Wasserstein distance), for a large class of dependent random variables, including for some dynamical systems. We first establish, in the second chapter, the moderate deviation principle, for non-adapted stationary sequences of bounded random variables with values in a Hilbert space, under martingale-type conditions. Applications to Cramér-Von Mises statistics, functions of linear processes (important in the study of forecasting problems), and stable Markov chains are given. The third chapter, we give a Central Limit Theorem for ergodic stationary sequences of martingale dierences in L^1. Then, by martingale approximation, we derive a Central Limit Theorem for some ergodic stationary sequences of L^1-valued random variables satisfying some projective criteria. This result allows us to get sufficient conditions to derive the asymptotic behavior of statistics of Wasserstein-type for a large class of dependent sequences. In particular, we give some applications to dynamical systems and causal linear processes. Finally, in order to construct asymptotic confidence intervals for the mean of a stationary sequence, we suggest a smoothed estimator of spectral density. In this last chapter, we give projective-conditions for the convergence in L^1 of a smoothed estimator of spectral density. Moreover, this result allows us via Central Limit Theorem, to get confidence regions for parameters in a parametric regression model.
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https://tel.archives-ouvertes.fr/tel-00440850
Contributor : Sophie Dede <>
Submitted on : Saturday, December 12, 2009 - 1:02:38 PM
Last modification on : Thursday, December 10, 2020 - 10:51:53 AM
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Sophie Dede. Théorèmes limites fonctionnels et estimation de la densité spectrale pour des suites stationnaires.. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2009. Français. ⟨tel-00440850⟩

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