Summary
Since the topology of weak convergence of probability distributions on the Borel σ-field of the space C= C([0, 1]) is metrizable, it is natural to describe the speed of convergence in weak functional limit theorems by means of an appropriate metric. Using the metric proposed by Prokhorov it is shown that under suitable conditions the rate of convergence in the functional central limit theorem for C-valued partial sum processes based on martingale difference arrays is the same as in the special case of row-wise independent random variables where this rate is known to be an optimal one.
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Haeusler, E. An exact rate of convergence in the functional central limit theorem for special martingale difference arrays. Z. Wahrscheinlichkeitstheorie verw Gebiete 65, 523–534 (1984). https://doi.org/10.1007/BF00531837
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DOI: https://doi.org/10.1007/BF00531837