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Speed of convergence in the central limit theorem for mixing Hilbert space valued random variables. (Vitesse de convergence dans le théorème central limite pour des variables aléatoires mélangeantes à valeurs dans un espace de Hilbert.) (French) Zbl 0557.60006

Summary: We consider random variables with values in a separable Hilbert space which constitute a \(\phi\)-mixing sequence or a strongly mixing field indexed by \(\mathbb Z^ d\). We show Marcinkiewicz-Zygmund inequalities and exponential inequalities where the random variables are bounded and mixing is geometric. We also show Berry-Esseen estimates for the central limit theorem in the stationary case; our estimates are of order \(n^{\eta -\gamma}\) with \(\gamma =1/4(2+d)\) for \(d\leq 2\), and \(\gamma =1/7d\) else, \(\eta\) is arbitrary little and n is the number of random variables.

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F05 Central limit and other weak theorems
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