Doukhan, Paul; Leon, José; Portal, Frédéric 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 C. R. Acad. Sci., Paris, Sér. I 298, 305-308 (1984). 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. Cited in 1 ReviewCited in 10 Documents MSC: 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60F05 Central limit and other weak theorems Keywords:strongly mixing field; Marcinkiewicz-Zygmund inequalities; exponential inequalities; Berry-Esseen estimates; central limit theorem PDFBibTeX XMLCite \textit{P. Doukhan} et al., C. R. Acad. Sci., Paris, Sér. I 298, 305--308 (1984; Zbl 0557.60006)