Summary
Very weak Bernoulli processes with values in a separable metric space are introduced. An estimate for the Prohorov distance in the central limit theorem is obtained. This estimate is used to establish a strong (almost sure) approximation of the partial sums of a very weak Bernoulli process by a Brownian motion where the error term is of the order O(t 1/2−γ). The proofs are based on a new version of the Berkes-Philipp approximation theorem.
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Eberlein, E. Strong approximation of very weak Bernoulli processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 62, 17–37 (1983). https://doi.org/10.1007/BF00532160
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DOI: https://doi.org/10.1007/BF00532160