Summary
We prove an estimate for the Prohorov-distance in the central limit theorem for strong mixing Banach space valued random variables. Using a recent variant of an approximation theorem of Berkes and Philipp (1979) we obtain as a corollary a strong invariance principle for absolutely regular sequences with error term \(t^{\tfrac{1}{2} - \gamma }\). For strong mixing sequences we prove a strong invariance principle with error term o((t log logt)1/2).
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Dehling, H. Limit theorems for sums of weakly dependent Banach space valued random variables. Z. Wahrscheinlichkeitstheorie verw Gebiete 63, 393–432 (1983). https://doi.org/10.1007/BF00542537
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DOI: https://doi.org/10.1007/BF00542537