# Théorèmes limites de la théorie des probabilités dans les systèmes dynamiques

Abstract : This thesis is devoted to limit theorems for strictly stationary sequences and random fields. We concentrate essentially on the central limit theorem and its invariance principle. First, we show with the help of a counter-example that for a strictly stationary absolutely regular sequence, the central limit theorem may hold but not the invariance principle. We also show that the central limit theorem does not take place for partial sums of a Hilbert space valued, strictly stationary and absolutely regular sequence, even if we assume that the normalized partial sums form a uniformly integrable family. Second, we investigate the Holderian invariance principle. We treat the case of $\tau$-dependent (Dedecker, Prieur, 2005) and $\rho$-mixing strictly stationary sequences. We provide a sufficient condition on the law of a strictly stationary martingale difference sequence and the quadratic variance which guarantee the invariance principle in a Hölder space. We construct a counter-example which shows its sharpness. We derive conditions in the spirit of Hannan (1979), and Maxwell and Woodroofe (2000) by a martingale approximation. We then discuss the martingale/coboundary decomposition. In dimension one, we provide sharp integrability conditions on the transfer function and the coboundary for which the later does not spoil the invariance principle, the law of the iterated logarithm or the strong law of large numbers if these theorems take place for the martingale involved in the decomposition. We also provide a sufficient condition for an orthomartingale/coboundary decomposition for strictly stationary random fields. Lastly, we establish tails inequalities for orthomartingale and Bernoulli random fields. We prove an invariance principle in Hölder spaces for these random fields using such inequalities.
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Contributor : Davide Giraudo <>
Submitted on : Friday, December 18, 2015 - 5:19:26 PM
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• HAL Id : tel-01246592, version 1

### Citation

Davide Giraudo. Théorèmes limites de la théorie des probabilités dans les systèmes dynamiques. Probabilités [math.PR]. Université de Rouen 2015. Français. ⟨tel-01246592⟩

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