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Principes d'invariance faible pour la mesure empirique d'une suite de variables aléatoires mélangeante
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  • Published: September 1987

Principes d'invariance faible pour la mesure empirique d'une suite de variables aléatoires mélangeante

  • P. Doukhan1,
  • J. R. Leon1 &
  • F. Portal1 

Probability Theory and Related Fields volume 76, pages 51–70 (1987)Cite this article

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  • 7 Citations

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Summary

We give weak invariance principles for the empirical measure of a stationary strongly mixing sequence \((\xi _k )_{k \geqq 0,} X_n (f) = \frac{1}{{\sqrt n }}\sum\limits_{k = 1}^n {(f(\xi _k ) - Ef(\xi _k ))} \). For the case where f∈B s ,the unit ball of the Sobolev space H s (X) of a riemannian compact manifold, and f is a Lipα function (\((\frac{1}{2} < \alpha \leqq {\text{1}})\)) we obtain logarithmic rates of convergence ɛ n such that, for a stationary sequence of gaussian processes Y n ,\(\mathbb{P}(\mathop {\sup }\limits_f |X_n (f) - Y_n (f)| \geqq \varepsilon _n {\text{)}} \leqq \varepsilon _n \). We also prove, for the case of kernel estimates \(\hat g_n \), the existence of a gaussian non stationary sequence of random processes (Y n (x)) x∈K indexed be a compact subset K of \(\mathbb{R}^{^d } \) and constants a, b, c>0 such that

$$\mathbb{P}{\text{(}}\mathop {{\text{sup}}}\limits_{{\text{x}} \in K} {\text{|(}}nh_n^d {\text{)}}^{{\text{1/2}}} (\hat g_n (x) - g(x)) - Y_n (x)| \geqq cn^{ - a} ) \leqq cn^{ - a} if h_n = n^{ - b} ;$$

finally we give estimates of the kind:

$$E\mathop {\sup }\limits_{x \in K} {\text{ }}(\hat g_n (x) - g(x))^2 \leqq cn^{ - a} {\text{ if }}h_n = n^{ - b} .$$

Here h n is the window of the kernel estimate \(\hat g_n \).

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Authors and Affiliations

  1. U.A. 743 «Statistique Appliquée» Mathématiques, Université Paris-Sud, Bât. 425, F-91405, Orsay, France

    P. Doukhan, J. R. Leon & F. Portal

Authors
  1. P. Doukhan
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  2. J. R. Leon
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  3. F. Portal
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Doukhan, P., Leon, J.R. & Portal, F. Principes d'invariance faible pour la mesure empirique d'une suite de variables aléatoires mélangeante. Probab. Th. Rel. Fields 76, 51–70 (1987). https://doi.org/10.1007/BF00390275

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  • Received: 18 March 1985

  • Revised: 06 October 1986

  • Issue Date: September 1987

  • DOI: https://doi.org/10.1007/BF00390275

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